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Volume of Cylinders

Subject: Mathematics

Age range: 14-16

Resource type: Lesson (complete)

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One FULL LESSON on finding the volume of cylinders. This lesson follows on from volume of prisms.

Contents of download:

• Clicker version : Normal PowerPoint lesson with which you can use a clicker / mouse / keyboard to continue animations and show solutions.
• Worksheets (including example and extension).

We are learning about: The volume of a cylinder We are learning to: Calculate the volume of a cylinder.

Differentiated objectives:

• Developing learners will be able to calculate the volume of a cylinder.
• Secure learners will be able to find a missing length in a cylinder given its volume.
• Excelling learners will be able to solve unfamiliar problems using their knowledge of calculating the volume of a cylinder.

Main: Walkthrough examples followed by practice questions on worksheets. Starts with basic calculating the volume moving on to finding missing lengths of a cylinder. All solutions given on PPT and in worksheet format.

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Mr Barton Maths Podcast

Long-form conversations about teaching and learning with craig barton, tes top 10 resources: perimeter, area and volume.

The following collection of resources have been assembled by the TES Maths Panel . They can be downloaded for free by registering on the TES website.

Perimeter, area and volume can be a fairly tedious topic to teach. On the other hand, it has the potential to be brought to life to capture students’ imagination and interest. Remembering formulae isn’t the way for students to learn this topic; it is through visual and interactive learning that it will stick.

Top 10 resources:

1. Circumference and perimeter treasure hunt

Age range:  11 – 16  Format:  PDF

An easy to use resource to challenge pupils. The compound shape questions are particularly useful.

2. Drawing, volume and surface area of prisms

Age range:  11 – 16  Format:  Word document

A functional resource that develops a variety of skills and comes with a plentiful list of extension tasks.

3. Area puzzle starter

An unusual resource to get students thinking at the start of a lesson.

4. Volume of revolution

Age range:  16 – 18  Format:  PDF

Learn how to use Autograph to teach volumes of revolution – an effective way to aid students’ understanding of 3D problems.

5. Area and perimeter of triangles

An engaging resource that requires students to find the area and perimeter of triangles – includes an extension task to consolidate understanding.

6. Finding the volume and surface area of a cuboid

Age range:  11 – 16  Format:  .swf

This resource provides help to students struggling to visualise cuboid related problems.

7. Pyramid volume proof

Age Range:  11 – 16  Format:  Web

A practical lesson that results in students deriving the formula for the volume of a pyramid.

8. Area and perimeter follow-me cards

This plenary activity offers a wide range of questions on a variety of shapes to challenge students’ understanding.

9. GCSE maths: Evaluating statements – length and area

Age range: 11 – 16  Format:  PDF

Challenge pupils and aid deeper understanding of conjectures with this series of statements and investigation task.

10. Area and perimeter PowerPoint 

Age range:  11 – 16  Format:  PowerPoint presentation

Visual and contextual ideas for creating or adapting a lesson on area and perimeter.

Colm Lynch,  maths secondary panel

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Volume Problem Solving

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To solve problems on this page, you should be familiar with the following: Volume - Cuboid Volume - Sphere Volume - Cylinder Volume - Pyramid

This wiki includes several problems motivated to enhance problem-solving skills. Before getting started, recall the following formulas:

• Volume of sphere with radius \(r:\) \( \frac43 \pi r^3 \)
• Volume of cube with side length \(L:\) \( L^3 \)
• Volume of cone with radius \(r\) and height \(h:\) \( \frac13\pi r^2h \)
• Volume of cylinder with radius \(r\) and height \(h:\) \( \pi r^2h\)
• Volume of a cuboid with length \(l\), breadth \(b\), and height \(h:\) \(lbh\)

Volume Problem Solving - Basic

Volume - problem solving - intermediate, volume problem solving - advanced.

This section revolves around the basic understanding of volume and using the formulas for finding the volume. A couple of examples are followed by several problems to try.

Find the volume of a cube of side length \(10\text{ cm}\). \[\begin{align} (\text {Volume of a cube}) & = {(\text {Side length}})^{3}\\ & = {10}^{3}\\ & = 1000 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

Find the volume of a cuboid of length \(10\text{ cm}\), breadth \(8\text{ cm}\). and height \(6\text{ cm}\). \[\begin{align} (\text {Area of a cuboid}) & = l × b × h\\ & = 10 × 8 × 6\\ & = 480 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

I made a large ice cream cone of a composite shape of a cone and a hemisphere. If the height of the cone is 10 and the diameter of both the cone and the hemisphere is 6, what is the volume of this ice cream cone? The volume of the composite figure is the sum of the volume of the cone and the volume of the hemisphere. Recall the formulas for the following two volumes: \( V_{\text{cone}} = \frac13 \pi r^2 h\) and \( V_{\text{sphere}} =\frac43 \pi r^3 \). Since the volume of a hemisphere is half the volume of a a sphere of the same radius, the total volume for this problem is \[\frac13 \pi r^2 h + \frac12 \cdot \frac43 \pi r^3. \] With height \(h =10\), and diameter \(d = 6\) or radius \(r = \frac d2 = 3 \), the total volume is \(48\pi. \ _\square \)

Find the volume of a cone having slant height \(17\text{ cm}\) and radius of the base \(15\text{ cm}\). Let \(h\) denote the height of the cone, then \[\begin{align} (\text{slant height}) &=\sqrt {h^2 + r^2}\\ 17&= \sqrt {h^2 + 15^2}\\ 289&= h^2 + 225\\ h^2&=64\\ h& = 8. \end{align}\] Since the formula for the volume of a cone is \(\dfrac {1}{3} ×\pi ×r^2×h\), the volume of the cone is \[ \frac {1}{3}×3.14× 225 × 8= 1884 ~\big(\text{cm}^{2}\big). \ _\square\]

Find the volume of the following figure which depicts a cone and an hemisphere, up to \(2\) decimal places. In this figure, the shape of the base of the cone is circular and the whole flat part of the hemisphere exactly coincides with the base of the cone (in other words, the base of the cone and the flat part of the hemisphere are the same). Use \(\pi=\frac{22}{7}.\) \[\begin{align} (\text{Volume of cone}) & = \dfrac {1}{3} \pi r^2 h\\ & = \dfrac {1 × 22 × 36 × 8}{3 × 7}\\ & = \dfrac {6336}{21} = 301.71 \\\\ (\text{Volume of hemisphere}) & = \dfrac {2}{3} \pi r^3\\ & = \dfrac {2 × 22 × 216}{3 × 7}\\ & = \dfrac {9504}{21} = 452.57 \\\\ (\text{Total volume of figure}) & = (301.71 + 452.57) \\ & = 754.28.\ _\square \end{align} \]

Try the following problems.

Find the volume (in \(\text{cm}^3\)) of a cube of side length \(5\text{ cm} \).

A spherical balloon is inflated until its volume becomes 27 times its original volume. Which of the following is true?

Bob has a pipe with a diameter of \(\frac { 6 }{ \sqrt { \pi } }\text{ cm} \) and a length of \(3\text{ m}\). How much water could be in this pipe at any one time, in \(\text{cm}^3?\)

What is the volume of the octahedron inside this \(8 \text{ in}^3\) cube?

A sector with radius \(10\text{ cm}\) and central angle \(45^\circ\) is to be made into a right circular cone. Find the volume of the cone.

\[\] Details and Assumptions:

• The arc length of the sector is equal to the circumference of the base of the cone.

Three identical tanks are shown above. The spheres in a given tank are the same size and packed wall-to-wall. If the tanks are filled to the top with water, then which tank would contain the most water?

A chocolate shop sells its products in 3 different shapes: a cylindrical bar, a spherical ball, and a cone. These 3 shapes are of the same height and radius, as shown in the picture. Which of these choices would give you the most chocolate?

\[\text{ I. A full cylindrical bar } \hspace{.4cm} \text{ or } \hspace{.45cm} \text{ II. A ball plus a cone }\]

How many cubes measuring 2 units on one side must be added to a cube measuring 8 units on one side to form a cube measuring 12 units on one side?

This section involves a deeper understanding of volume and the formulas to find the volume. Here are a couple of worked out examples followed by several "Try It Yourself" problems:

\(12\) spheres of the same size are made from melting a solid cylinder of \(16\text{ cm}\) diameter and \(2\text{ cm}\) height. Find the diameter of each sphere. Use \(\pi=\frac{22}{7}.\) The volume of the cylinder is \[\pi× r^2 × h = \frac {22×8^2×2}{7}= \frac {2816}{7}.\] Let the radius of each sphere be \(r\text{ cm}.\) Then the volume of each sphere in \(\text{cm}^3\) is \[\dfrac {4×22×r^3}{3×7} = \dfrac{88×r^3}{21}.\] Since the number of spheres is \(\frac {\text{Volume of cylinder}}{\text {Volume of 1 sphere}},\) \[\begin{align} 12 &= \dfrac{2816×21}{7×88×r^3}\\ &= \dfrac {96}{r^3}\\ r^3 &= \dfrac {96}{12}\\ &= 8\\ \Rightarrow r &= 2. \end{align}\] Therefore, the diameter of each sphere is \[2\times r = 2\times 2 = 4 ~(\text{cm}). \ _\square\]

Find the volume of a hemispherical shell whose outer radius is \(7\text{ cm}\) and inner radius is \(3\text{ cm}\), up to \(2\) decimal places. We have \[\begin{align} (\text {Volume of inner hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × R^3\\ & = \dfrac {1 × 4 × 22 × 27}{2 × 3 × 7}\\ & = \dfrac {396}{7}\\ & = 56.57 ~\big(\text{cm}^{3}\big) \\\\ (\text {Volume of outer hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × r^3\\ & = \dfrac {1 × 4 × 22 × 343}{2 × 3 × 7}\\ & = \dfrac {2156}{7}\\ & = 718.66 ~\big(\text{cm}^{3}\big) \\\\ (\text{Volume of hemispherical shell}) & = (\text{V. of outer hemisphere}) - (\text{V. of inner hemisphere})\\ & = 718.66 - 56.57 \\ & = 662.09 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

A student did an experiment using a cone, a sphere, and a cylinder each having the same radius and height. He started with the cylinder full of liquid and then poured it into the cone until the cone was full. Then, he began pouring the remaining liquid from the cylinder into the sphere. What was the result which he observed?

There are two identical right circular cones each of height \(2\text{ cm}.\) They are placed vertically, with their apex pointing downwards, and one cone is vertically above the other. At the start, the upper cone is full of water and the lower cone is empty.

Water drips down through a hole in the apex of the upper cone into the lower cone. When the height of water in the upper cone is \(1\text{ cm},\) what is the height of water in the lower cone (in \(\text{cm}\))?

On each face of a cuboid, the sum of its perimeter and its area is written. The numbers recorded this way are 16, 24, and 31, each written on a pair of opposite sides of the cuboid. The volume of the cuboid lies between \(\text{__________}.\)

A cube rests inside a sphere such that each vertex touches the sphere. The radius of the sphere is \(6 \text{ cm}.\) Determine the volume of the cube.

If the volume of the cube can be expressed in the form of \(a\sqrt{3} \text{ cm}^{3}\), find the value of \(a\).

A sphere has volume \(x \text{ m}^3 \) and surface area \(x \text{ m}^2 \). Keeping its diameter as body diagonal, a cube is made which has volume \(a \text{ m}^3 \) and surface area \(b \text{ m}^2 \). What is the ratio \(a:b?\)

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction \( \frac{m}{n} \), for relatively prime integers \(m\) and \(n\). Compute \(m+n\).

The square-based pyramid A is inscribed within a cube while the tetrahedral pyramid B has its sides equal to the square's diagonal (red) as shown.

Which pyramid has more volume?

Please remember this section contains highly advanced problems of volume. Here it goes:

Cube \(ABCDEFGH\), labeled as shown above, has edge length \(1\) and is cut by a plane passing through vertex \(D\) and the midpoints \(M\) and \(N\) of \(\overline{AB}\) and \(\overline{CG}\) respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

If the American NFL regulation football

has a tip-to-tip length of \(11\) inches and a largest round circumference of \(22\) in the middle, then the volume of the American football is \(\text{____________}.\)

Note: The American NFL regulation football is not an ellipsoid. The long cross-section consists of two circular arcs meeting at the tips. Don't use the volume formula for an ellipsoid.

Answer is in cubic inches.

Consider a solid formed by the intersection of three orthogonal cylinders, each of diameter \( D = 10 \).

What is the volume of this solid?

Consider a tetrahedron with side lengths \(2, 3, 3, 4, 5, 5\). The largest possible volume of this tetrahedron has the form \( \frac {a \sqrt{b}}{c}\), where \(b\) is an integer that's not divisible by the square of any prime, \(a\) and \(c\) are positive, coprime integers. What is the value of \(a+b+c\)?

Let there be a solid characterized by the equation \[{ \left( \frac { x }{ a } \right) }^{ 2.5 }+{ \left( \frac { y }{ b } \right) }^{ 2.5 } + { \left( \frac { z }{ c } \right) }^{ 2.5 }<1.\]

Calculate the volume of this solid if \(a = b =2\) and \(c = 3\).

• Surface Area

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Volume of Prisms

Solid geometry is concerned with three-dimensional shapes. In these lessons, we will learn

• what is a prism?
• how to find the volume of prisms.
• how to solve word problems about prisms.

Related Pages Volume Formula Volume Of Prism Worksheet Volume Formulas Explained More Geometry Lessons

A prism is a solid that has two parallel faces which are congruent polygons at both ends. These faces form the bases of the prism. A prism is named after the shape of its base.

The other faces are in the shape of parallelograms. They are called lateral faces.

A right prism is a prism that has its bases perpendicular to its lateral surfaces. If the bases are not perpendicular to its lateral bases then it is called an oblique prism.

When we cut a prism parallel to the base, we get a cross section of a prism. The cross section has the same size and shape as the base.

What is a prism and distinguishes between a right prism and an oblique prism?

How to label the parts of a prism and how to distinguish between an oblique and a right prism?

Volume of a Prism

The volume of a right prism is given by the formula:

Volume = Area of base × height = Ah

where A is the area of the base and h is the height or length of the prism.

Worksheet to calculate volume of prisms and pyramids.

Example: Find the volume of the following right prism.

Solution: Volume = Ah = 25 cm 2 × 9 cm = 225 cm 3

Example: Find the volume of the following right prism

Solution: First, we need to calculate the area of the triangular base.

We would need to use Pythagorean theorem to calculate the height of the triangle.

h 2 + 3 2 = 5 2

Volume of prism = Ah = 12 cm 2 × 8 cm = 96 cm 3

How to find the volume of a rectangular and a triangular prism? Step 1: Find the area of the base. Step 2: Multiply the area of the base times the height.

How to find the volume of any prism, right or oblique using a general formula?

Word problems about volume of prisms

The following video shows how to solve a word problem involving the volume of prisms.

Example: Find the volume and capacity of a swimming pool which is made up of a rectangular and trapezoidal prism.

Use the given net to determine the surface area and volume of a triangular prism

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Course: 5th grade   >   Unit 11

• Volume word problem: water tank

Volume word problems

• Volume of rectangular prisms review
• Volume: FAQ
• (Choice A)   8  cm ‍   long, 1  cm ‍   wide, 3  cm ‍   high A 8  cm ‍   long, 1  cm ‍   wide, 3  cm ‍   high
• (Choice B)   10  cm ‍   long, 4  cm ‍   wide, 10  cm ‍   high B 10  cm ‍   long, 4  cm ‍   wide, 10  cm ‍   high
• (Choice C)   2  cm ‍   long, 2  cm ‍   wide, 6  cm ‍   high C 2  cm ‍   long, 2  cm ‍   wide, 6  cm ‍   high

Resources you can trust

Problem solving with volumes of prisms

A problem-solving activity using volumes of prisms and cylinders, plus constructing and solving simple equations.

Students are shown prisms and a cylinder with measurements given either numerically or as unknowns x, y and z.

They must use the information to form and solve simple equations to find all the unknowns as well as the radius of the cylinder.

Students can then create their own prism with the same volume to go in the empty box.

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Perimeter, Area and Volume Short Problems

This is part of our collection of Short Problems . You may also be interested in our longer problems on Perimeter, Area and Volume Age 11-14 and Age 14-16 . Printable worksheets containing selections of these problems are available here:

Cuboid Faces

Can you find the volume of a cuboid, given the areas of its faces?

Chequered Cuboid

A large cuboid is made from cubes of equal size. What fraction of the surface area of the large cuboid is black?

Star in a Hexagon

Weekly Problem 20 - 2017 The diagram shows a design formed by drawing six lines in a regular hexagon. What fraction of the hexagon is shaded?

Which of the following shaded regions has an area different from the other shaded regions?

Mid-point Area

M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?

Six Circles

In the diagram, six circles of equal size touch adjacent circles and the sides of the large rectangle. What is the perimeter of the large rectangle?

Which of the areas shown in the hexagons are equal to each other?

Nested Squares

What proportion of the diagram is shaded?

Giant Rubik's Cube

How many cubes would be visible in a 12 by 12 by 12 Rubik’s cube?

Squares in a Square

In the diagram, the small squares are all the same size. What fraction of the large square is shaded?

Sideways Ratio

Weekly Problem 33 - 2014 A rectangle with area $125\text{cm}^2$ has sides in the ratio $4:5$. What is the perimeter of the rectangle?

Dividing a Square

A square is divided into three shapes which all have equal areas. Can you find the length of this side?

Cubic Masterpiece

Weekly Problem 49 - 2014 A blue cube is cut into 27 smaller cubes of equal size. What fraction of the total surface area of these cubes is blue?

Scalene Area

Can you find the area of this scalene triangle?

Double Cover

Weekly Problem 3 - 2007 What is the ratio of the area of the table covered twice, to the uncovered area?

Semicircle Shape

Find the shaded area of these shapes with perimeters made of semicircles.

Weekly Problem 9 - 2006 What fraction of the area of the rectangle is shaded?

Tangram Area

Weekly Problem 29 - 2008 The seven pieces in this 12 cm by 12 cm square make a Tangram set. What is the area of the shaded parallelogram?

Tile Border

What is the largest possible number of yellow tiles in this pattern?

If the area of a face of a cuboid is one quarter of the area of each of the other two visible faces, what is the area of these faces?

Open the Box

Weekly Problem 37 - 2015 A piece of card is folded to make an open box. Given its surface area, can you work out its volume?

Triangles' Triangle

Weekly Problem 27 - 2009 The perimeter of a large triangle is 24 cm. What is the total length of the black lines used to draw the figure?

Strawberries and Peas

Weekly Problem 17 - 2017 Yasmin lengthened one side of her pea bed by 3m to make it a square. This reduced her strawberry patch by $15m^2$. What was the original area of her pea bed?

Tilted Tank

Can you find the height of the water in this tilted tank when it is flat?

Cubes on a Cube

Weekly Problem 26 - 2016 A cube has each of its faces covered by one face of an identical cube, making a solid as shown. What is the surface area of the solid?

Packing Small Boxes

How many small boxes will fit inside the big box?

Hawaiian Earring

What fraction of the larger circle is outside the smaller circle?

Weekly Problem 23 - 2017 Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle. What is the side length of the small triangles?

Exactly Three-quarters

Weekly Problem 30 - 2007 Three-quarters of the area of the rectangle has been shaded. What is the length of x?

Weekly Problem 28 - 2006 What can you say about the rectangles that form this L-shape?

Leaning Over

Weekly Problem 31 - 2017 The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?

Line of Squares

Weekly Problem 35 - 2016 What is the total perimeter of the squares, if the line GH in the diagram is 24cm?

Intersecting Squares

Weekly Problem 32 - 2014 Three overlapping squares are shown. If you know the areas of the overlapping and non-overlapping parts, can you work out the side lengths of the squares?

Christmas Cut-out

Weekly Problem 52 - 2017 Sue cuts some squares from a piece of paper to make a Christmas decoration. What is the perimeter of the resulting shape?

Weekly Problem 16 - 2008 A 30cm x 40cm page of a book includes a 2cm margin on each side... What percentage of the page is occupied by the margins?

Triangle in a Hexagon

Weekly Problem 3 - 2009 What fraction of the area of this regular hexagon is the shaded triangle?

Rectangle Split

Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.

Pile Driver

Weekly Problem 38 - 2015 Where does the line through P that halves the figure shown meet the edge XY?

Four Square

Weekly Problem 17 - 2015 A square contains two overlapping squares. What is the total of the shaded regions?

Hexagon Slices

Weekly Problem 37 - 2007 This regular hexagon has been divided into four trapezia and one hexagon.... what is the ratio of the lengths of sides p, q and r?

Semicircular Design

Weekly Problem 9 - 2016 The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

The circle of radius 4cm is divided into four congruent parts by arcs of radius 2cm as shown. What is the length of the perimeter of one of the parts, in cm?

Annulus Area

Weekly Problem 38 - 2011 Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?

What is the area of the shape enclosed by the line and arcs?

Semicircle Stack

What is the total area enclosed by the three semicicles?

Bobbly Perimeter

Find the perimeter of this shape made of semicircles

Ratio of Areas

What is the ratio of the area of the hexagon to the area of the triangle?

Penny Farthing

Boris' bicycle has a smaller back wheel than front wheel. Can you work out how many revolutions the front wheel made if the back wheel did 120,000?

Four semicircles are drawn on a line to form a shape. What is the area of this shape?

Crazy Shading

Can you work out the fraction of the larger square that is covered by the shaded area?

Circle in a Semicircle

Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

Which of these two paths made of semicircles is shorter?

Square Ratio

A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?

Similar Cylinders

Two similar cylinders are formed from a block of metal. What is the volume of the smaller cylinder?

Circled Corners

Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?

Square Flower

The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?

Pencil Turning

Rotating a pencil twice about two different points gives surprising results...

Wood Pile Perimeter

Weekly Problem 30 - 2011 Three touching circles have an interesting area between them...

Semicircle Distance

Can you find the shortest distance between the semicircles given the area between them?

Trisected Triangle

Weekly Problem 34 - 2015 Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

Rolling Inside

Weekly Problem 11 - 2007 A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression for the number of revolutions the circle makes.

Square in a Circle in a Square

What is the ratio of the areas of the squares in the diagram?

Maximised Area

Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

Four Leaf Clover

The diagram shows four equal discs and a square. What is the perimeter of the figure?

The diagram shows a shaded shape bounded by circular arcs. What is the difference in area betweeen this and the equilateral triangle shown?

Canny Fraction

What fraction of the volume of this can is filled with lemonade?

Loo Roll Emergency

When the roll of toilet paper is half as wide, what percentage of the paper is left?

Rectangle Cutting

Tom and Jerry start with identical sheets of paper. Each one cuts his sheet in a different way. Can you find the perimeter of the original sheet?

Cones and Spheres

A solid metal cone is melted down and turned into spheres. How many spheres can be made?

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

Half an Annulus

Can you locate the point on an annulus that splits it into two areas?

Candy Floss

What length of candy floss can Rita spin from her cylinder of sugar?

Sticky Tape

Work out the radius of a roll of adhesive tape.

Emptied Cube

Weekly Problem 26 - 2015 What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?

Weekly Problem 5 - 2006 How many times does the inside disc have to roll around the inside of the ring to return to its initial position?

Cutting a rectangle from a corner to a point on the opposite side splits its area in the ratio 1:2. What is the ratio of a:b?

Weekly Problem 52 - 2014 Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

Tilted Aquarium

Can you find the depth of water in this aquarium?

Running Race

Weekly Problem 13 - 2006 If three runners run at the same constant speed around the race tracks, in which order do they finish?

Can you find the area of the yellow part of this snake's eye?

Cut-up Square

Weekly Problem 15 - 2015 In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?

Sinking Feeling

Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?

Weekly Problem 51 - 2015 Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?

Can you work out the shaded area surrounded by these arcs?

Resourceaholic

Ideas and resources for teaching secondary school mathematics

• Blog Archive

• Time  - Boss Maths
• Converting time odd one out  ( answers ) - Dr Austin Maths
• KS2 Time  - Maths4Everyone
• Time worksheets  - teachingimage.com 
• Converting decimals to hours and minutes  - @mathsmrgordon via variationtheory.com
• Reading scales  - SMILE
• Reading and interpreting scales  - fionajones88 on TES
• Using scale diagrams and maps  - Boss Maths
• Scale drawings  - Boss Maths
• Interpreting Map Scales Practice Grid ( Answers ) - Dr Austin Maths
• Using Map Scales Practice Grid ( Answers ) - Dr Austin Maths
• Scale Drawing and Maps - dannytheref on TES
• Scale Drawing - CIMT

Scale Drawings Practice Grid ( answers ) - Dr Austin Maths

• Scale drawings - Dr Austin Maths
• DIY Units  - Andy Lutwyche on TES
• Identifying measurements - TeachIt Maths
• Metric/imperial conversion bursts - TeachIt Maths
• Metric units table  - The Chalk Face
• Metric units activity  - The Chalk Face
• Metric and imperial units:  activity 1  &  activity 2  - The Chalk Face
• Convert lengths bingo cards/card match  - Nuffield Foundation
• Units and large numbers  - Teachit Maths
• Units of Measurement - CIMT
• History of the metric system - Dan Walker on TES
• Why the metric system matters - Matt Anticole on Youtube
• Converting between metric units of measures of length and mass  - Boss Maths
• Metric unit conversions: lengths - Josh Cutts via variationtheory.com
• Metric unit conversions quiz - by me!
• Converting between metric units of measures of area and volume  - Boss Maths
• Money  - Boss Maths
• Universcale  (magnitude exploration tool) - Nikon

[ back to top ]

• Loci and regions - Median Don Steward
• Loci challenges booklet - The Chalk Face
• Loci investigations - The Chalk Face
• Loci matching exercise - @GiftedBA
• Mixed loci problems lesson - Boss Maths
• Grid Loci - Median Don Steward
• Loci presentation and worksheets -  @DJUdall 
• Festival Safety - idearefresh.co.uk
• Loci practice grid  ( answers ) - Dr Austin Maths
• Loci exam questions
• Constructing perpendicular bisectors, angle bisectors, and perpendiculars to or from a point lesson - Boss Maths
• Constructing triangles lesson - Boss Maths
• Constructing triangles lesson - cparkinson3 on TES
• Constructing perpendicular and angle bisectors lesson - cparkinson3 on TES
• Construct a scenario - Teachit Maths
• Constructions tasks - Median Don Steward
• Contructions - Dan Walker on TES
• Online constructions demonstrations and tools
• Construction animated gifs  (to paste into PowerPoints)
• Interactive Construction Tool - MathsPad.co.uk
• Euclid the Game
• Speed, distance, time task - Segar Rogers
• Speed activity - The Chalk Face
• Using compound units - Boss Maths
• Finding speed  - @fortyninecubed via variationtheory.com 
• Speed, distance, time questions  ( answers ) - @taylorda01
• Speed, distance, time questions II  ( answers ) - @taylorda01
• Speeding - Median Don Steward
• Speed calculations fill in the blanks  ( answers ) - Dr Austin Maths
• Harder speed calculations fill in the blanks  ( answers ) - Dr Austin Maths
• Distance-Time Graphs lesson - cparkinson3 on TES
• Distance-time graphs step-by-step worksheet - Labrown20 on TES
• Distance-time graphs and average speed  - Median Don Steward
• Interpreting Distance-Time Graphs  - Maths Assessment Project (activities at the back)
• Distance Time Graphs - mathshko
• Journey - Maths Assessment Project (activities at the back)
• Hurdles Race - Transum
• Super hero activity  - @jase_wanner 
• Compound Measures - Median Don Steward
• Density, mass, volume questions  ( answers ) - @taylorda01
• Exam question practice - density - Maths4Everyone on TES
• Density questions from Linked Pair GCSE - compiled by me!
• Identifying similar triangles  - Mathematics Assessment Project. 
• Similar shapes booklet  - tumshy on TES.
• Similarity  - CIMT
• Similar triangles - steele1989 on TES
• Similar Triangles notes and exercises - martahidegkuti.com
• Similar shapes odd one out ( answers ) - Dr Austin Maths
• Similar Shapes Practice Strips ( Answers ) - Dr Austin Maths
• Similar Shapes Crack the Code ( Answers ) - Dr Austin Maths
• Similar triangles questions  ( Answers ) - @taylorda01
• Similarity slides - Don Steward
• Similar triangles exam style questions ( answers ) - 1st Class Maths
• Similar shapes exam style questions  ( answers ) - 1st Class Maths
• Model Boat  - The Chalk Face 
• Similar shapes and equations - Mr Thompson on TES
• Evaluating Statements about Enlargements  - Mathematics Assessment Project
• Length, Area and Volume Factors - Median Don Steward
• Lengths, areas and volumes in similar shapes - Boss Maths
• Similar Areas and Volumes Fill In The Blanks ( Answers ) - Dr Austin Maths
• Similar Areas and Volumes Practice Strips ( Answers ) - Dr Austin Maths
• Harder Similar Areas and Volumes Fill In The Blanks ( Answers ) - Dr Austin Maths
• Harder Similar Areas and Volumes Practice Strips ( Answers ) - Dr Austin Maths
• Similar area and volume - mathshelper.co.uk
• Similar shapes GCSE revision - Maths4Everyone on TES
• Baby surface area  - The Chalk Face
• Similarity Check In - OCR
• Similar shapes (area and volume) exam questions ( answers ) - Maths Genie
• Prove it! - MathsPad
• Similar shapes full coverage GCSE questions  - compiled by Dr Frost
• Similar area and volume exam style questions ( answers ) - 1st Class Maths
• Congruence criteria for triangles (SSS, SAS, ASA, RHS)  - Boss Maths
• Congruent triangles worksheet - mathsmalakiss
• Analysing congruency proofs - Mathematics Assessment Project
• Congruent triangles
• Complete the congruence proof  - topdrawer.aamt.edu.au
• Constructions and congruence in triangles - Median Don Steward
• Congruence and similarity assessment - topdrawer.aamt.edu.au
• Congruence Check In - OCR
• Identify and Label Angles and Lengths  - fionaryan88 on TES
• Conventions for labelling the sides and angles of triangles  - Boss Maths
• Lines and angles naming and vocabulary activity  - alicecreswick on TES
• Drawing diagrams from a written description  - Boss Maths
• Understanding Angle Labels - Dan Draper
• Estimating and naming angles  - MathsPad
• CIMT angles: KS3 and KS4 - CIMT via TES
• Measuring line segments and angles in geometric figures - Boss Maths
• Measuring angles - demo and worksheets  - GoTeachMaths
• Introducing angles  - Gareth Evans on TES
• Basic angle properties activity  - Mark Horley
• Angle rules crack the code  ( answers ) - Dr Austin Maths
• Angles on a straight line worksheets  - Maths4Everyone on TES
• Angles around a point worksheets  - Maths4Everyone on TES
• Angles at a point, angles at a point on a straight line, vertically opposite angles  - Boss Maths
• Angles in triangles (worksheet bundle) - Maths4Everyone on TES
• Angles in a triangle extra practice  - Maths4Everyone on TES
• Isosceles triangle angles  - Median Don Steward
• Angles practice makes perfect - Median Don Steward
• Angles with algebra - Don Steward
• Angle facts questions  ( answers & supporting material ) - @taylorda01
• Angles in triangles and quadrilateral questions  ( answers ) - @taylorda01
• Missing angles  - Median Don Steward
• Two isosceles triangles stuck together - Median Don Steward
• Isosceles triangle proofs  - Median Don Steward
• Angles in triangles - equations  - MissBrookesMaths
• Angles in quadrilaterals - equations  - MissBrookesMaths
• Angle chases  - thanks to @MathedUp
• Angles and ratio  - Sam Blatherwick
• Angle problems - Nrich 
• Mixed angle puzzles - collated by me, see sheet for original sources
• There are loads of brilliant angles activities from  MathsPad  for a small subscription
• Alternate and corresponding angles on parallel lines - Boss Maths
• Spot the Angle  - MathsPad
• Interactive Parallel Lines Tool  - MathsPad
• Angles in parallel lines questions   ( answers  &  supporting material ) - @taylorda01
• Angles in parallel lines tick or trash  - MissBrookesMaths
• Unmarked angles  - SMILE
• Missing angles  - SMILE
• Angles in Parallel Lines (Worksheet Bundle)  - Maths4Everyone on TES
• Angles in parallel lines practice grids ( Answers ) - Dr Austin Maths
• Angles in Parallel Lines GCSE questions - Maths4Everyone
• Parallel line angles  - Median Don Steward
• Angles in parallel lines: comparing methods  - by me!
• Angles in Parallel Lines - Solving Linear Equations  - Mr Thompson on TES
• Solving linear equations from parallel lines - Nathan Day
• Parallelogram mazes - mathymcmatherson
• Parallel line angle relationships and proof  - Median Don Steward
• Angles in parallel lines GCSE questions  - Labrown20 on TES
• Death Star Angles  - dooranran on TES
• Mixed puzzles  -  collated by me, see sheet for original sources
• Maths GCSE Bearings  - collinbillet on TES
• Bearings lesson - Boss Maths
• Angles, bearings and maps - CIMT
• Bearings teaching ideas and activities - Colin Foster
• Angle properties and bearings - pas1001 on TES
• Bearings worksheets  and Finding bearings - The Chalk Face
• Calculating bearings - Richard Tock
• Calculating bearings - Maths4Everyone on TES
• Measuring bearings match up ( answers ) - Dr Austin Maths
• Calculating bearings practice grid ( answers ) - Dr Austin Maths
• Tour de France bearings activity  ( answers ) - Dr Austin Maths
• Find the Treasure - MathsPad
• Reasoning with Bearings - Dan Draper
• Bearings - Median Don Steward
• GCSE 9-1 Exam Practice Questions (Bearings)  - Maths4Everyone on TES

• Exterior angles tool - MathsPad
• Identifying Exterior Angles  - jamesgreenland on TES
• Angle sums investigation - MathsPad
• Angle in polygons  - The Chalk Face
• Applying angle theorems  - Mathematics Assessment Project
• Angles practice makes perfect  - Median Don Steward
• Angles homework  - collated by me! Questions from mathsmalakiss.com
• Angles in polygons  ( solutions ) - CazoomMaths
• Angles in Regular Polygons Practice Grid ( Answers ) - Dr Austin Maths
• Angles in Irregular Polygons ( Answers ) - Dr Austin Maths
• Exterior angles of convex polygons - Mr Draper Maths
• Using ratio to find angles in polygons - Stephen Gregory
• Angle Chasing II - Polygons and Parallel Lines - ItsMattKennedy on TES
• Angles in Polygons Multi Step Challenges - Maths4Everyone
• Angles in polygons GCSE revision - Maths4Everyone on TES
• Regular polygon angle compilation  and Pentagon angle compilation  - Median Don Steward
• Flowers - Median Don Steward
• Polygon Pile Up - Jon Orr
• Stained glass tessellations  - Teachit Maths
• Surrounding a point   - Median Don Steward

• 2D geometry – terms and notation - Boss Maths
• Properties of special triangles and quadrilaterals - Boss Maths
• Shape properties logic puzzle  - Teachit Maths
• Special Quadrilaterals - CIMT
• Describing and defining quadrilaterals - Mathematics Assessment Project
• Quadrilateral family tree
• Quadrilateral flowchart puzzle  - topdrawer.aamt.edu.au
• Quadrilateral Properties Quiz  - topdrawer.aamt.edu.au
• Quadrilateral property match-up  - topdrawer.aamt.edu.au
• Quadrilaterals Always Sometimes Never  - Lisa Bejarano
• Special quadrilaterals  - Teachit Maths
• What quadrilateral am I?   - Eddie Woo
• Polygon Properties  - Transum
• Shape Shoot  - Flash Maths
• Draw the shape - Don Steward
• Shapes and their Descriptions  - Shahira Ibrahim on TES
• Circle theorems on Geogebra - Michael Borcherds
• Geogebra apps and worksheet  - themathsteacher.com
• Circle theorems - including Geogebra applets - Boss Maths
• Introducing Circle Theorems - Nathan Day
• Circle theorems - first steps - Maths4Everyone on TES
• Blank circle theorems - The Chalk Face
• Circle Theorems Practice Grids - Dr Austin Maths
• Circle Theorems misconceptions tasks  - mathshko.com
• Circle theorems worksheet  - MathsPad
• Circle Theorems Reasoning Task - @rodrigotweets1 (with questions from Don Steward)
• Great angle chase  - topdrawer.aamt.edu.au
• Circle theorems codebreaker - alutwyche on TES
• Circle theorems problems - Maths Malakiss
• Solving equations with circle theorems  - Nathan Day
• Solving further equations with circle theorems - Karen Hancock
• Circle theorems revision exercise  - keyboardmonkey on TES
• Circle theorems meet 0.5absinC  - Median Don Steward
• Don Steward's circle theorems presentations  
• Circle Theorems Exam Style Questions ( answers ) - 1st Class Maths
• Circle Theorem Proofs Exam Style Questions ( answers ) - 1st Class Maths
• See my post  Ideas for Teaching Circle Theorems for more ideas and resources
• Area and Perimeter Lessons and Resources - MEI via Oak National Academy
• Right angled triangle areas - SMILE
• Counting Squares  - MathsPad
• Areas of Polygons  - SMILE
• Polygon Areas - SMILE
• Practical perimeters - TeachIt Maths
• Perimeter Overlearning - Dan Draper
• Rectangle areas - Don Steward
• Area of a triangle  - Kyle Gillies vis Starting Point Maths
• Triangle areas, various ways - Median Don Steward
• Area tasks - @giftedHKO
• Area of a parallelogram  - Kyle Gillies vis Starting Point Maths
• Area of Parallelograms and Trapeziums Match-Up  ( answers ) - Dr Austin Maths
• Area of a parallelogram and algebra - Mr Thompson on TES
• Area of parallelogram problem solving - Mr Thompson on TES
• Areas of trapeziums interwoven - Karen Hancock
• Area of trapezium - finding a missing length - Mr Thompson on TES
• Blog post: Thinking About Areas of Parallelograms - Paul Rowlandson
• Perimeter questions  ( answers ) - @taylorda01
• Area of 2D shapes questions ( answers ) - @taylorda01
• Area of 2D shapes questions II  ( answers ) - @taylorda01
• Area of 2D shapes Crack the Code  ( answers ) - Dr Austin Maths
• Perimeter and Area Tasks - John Mason
• Area and Perimeter Match - Don Steward
• Area practice - TeachIt Maths
• Examples and exercises from BossMaths: 
• Area of a rectangle
• Area of a triangle
• Area of a parallelogram
• Area of a trapezium
• Perimeter of polygons
• Perimeter and area of composite shapes made up of polygons .
• Area of 2D shapes review - TeachIt Maths
• Areas of Flags  - Owen134866 on TES
• L shapes (slides and worksheet) - MathsPad
• Area of L shaped diagrams - TeachIt Maths
• L shapes with fractions - Nathan Day
• Rectilinear areas - TeachIt Maths
• L shaped perimeters  - Don Steward
• Border areas - TeachIt Maths
• Compound Shapes Matching Cards  - Nuffield Foundation
• Compound rectangular shapes  - Median Don Steward 
• Equable shapes - Median Don Steward
• Area challenge - TeachIt Maths
• Area mazes - Don Steward
• Area puzzles - via STEM Centre

• Rhymes for the area and circumference of a circle
• Circles - introducing circumference - Median Don Steward 
• Circumference of a circle lesson - cparkinson3 on TES
• Circles questions - Teacher Resources Online
• Blog post: Thinking About Calculating Areas of Circles  - Paul Rowlandson
• Pi and the circumference of circles - Maths4Everyone on TES
• Non calculator area of a circle - Mr Thompson on TES
• Areas of a circle lesson - cparkinson3 on TES
• Area and circumference of a circle lesson - cparkinson3 on TES
• Area of a circle - Maths4Everyone on TES
• Circumference and Area - TeachIt Maths
• Area of a circle
• Circumference of a circle
• Perimeter and area of composite shapes made up of polygons and sectors of circles
• Area of a circle questions  ( answers ) - @taylorda01
• Mensuration - Teacher Resources Online
• Penny Farthing - The Chalk Face
• Geogebra showing revolution is circumference   - Tim Brzezinski
• Mixed circles questions with answers - collated by me from various sources
• Areas of Flags with Circles  - Owen134866 on TES
• Eight Circles - illustrativemathematics.org
• Area and perimeter of a sector lesson - cparkinson3 on TES
• Arc length and sector area questions  ( answers ) - @taylorda01
• Arc length and sector area - Median Don Steward
• Arcs and sectors worksheet - source unknown (sorry!)
• Area and perimeter of compound shapes  - mrwhy1089 on TES 
• Angle and radius of a sector lesson - cparkinson3 on TES
• Sectors Fill in the Gaps - by me!
• Circle sector problems - Access Maths
• Paper clip - Illustrative Mathematics
• Concentric circular rings - Median Don Steward
• Segment areas - mathshelper.co.uk
• Sectors GCSE Exam Practice  - Maths4Everyone on TES
• Areas of sectors GCSE revision - Maths4Everyone on TES
• Area of shaded regions GCSE revision - Maths4Everyone on TES
• Sectors, arcs and perimeters GCSE revision - Maths4Everyone on TES
• Basic cuboid volume - SMILE
• Interactive cuboid  - The Chalk Face
• Volume of a Cuboid Practice Grid  ( Answers ) - Dr Austin Maths
• Volume of a Cuboid Challenge Activity ( Answers ) - Dr Austin Maths
• Volume of Cubes and Cuboids Match-Up ( Answers ) - Dr Austin Maths
• Cuboid surface area - Median Don Steward
• Surface area of cuboids - problem solving - Mr Thompson on TES
• Cuboid volume and surface area - Median Don Steward
• Volume of prism lessons  - cparkinson3 on TES
• Volume of a Prism Practice Grid  ( Answers ) - Dr Austin Maths
• Volume and Surface Area of Prisms Crack the Code ( Answers ) - Dr Austin Maths
• Surface area of prism lesson  - cparkinson3 on TES
• Triangular prisms  - Median Don Steward
• Cornflakes problem  - The Chalk Face
• Fuel tank  - The Chalk Face
• Cylinder volume questions - Median Don Steward
• Volume of a Cylinder Practice Grid ( Answers ) - Dr Austin Maths
• Problem solving with volumes of prisms  - Teachit Maths
• Harder surface area questions  - Median Don Steward
• Sphere volume - Median Don Steward
• Golden balls  - The Chalk Face
• Volume of a cone questions  ( answers ) - @taylorda01
• Cone Volume - Median Don Steward
• Volume of a frustum - BJHarrison1 on TES
• Investigating the surface area of a cone  - Maths Sandpit
• Surface area of a cone questions  ( answers ) - @taylorda01
• Cone Surface Area - Median Don Steward
• Volume and area GCSE questions  - collated by me
• Volume GCSE questions  - UKMaths on TES
• Volume and Surface Area Revision Practice Grid  ( Answers ) - Dr Austin Maths
• Backwards Volumes - Starting Point Maths
• Volume of solids questions  ( answers ) - @taylorda01
• Surface area of solids questions  ( answers ) - @taylorda01
• Finding the volume of compound objects  - Shell Centre
• Functional Volume Questions - Access Maths
• Volume and Surface Area Revision Carousel - alisongilroy on TES
• Spheres, Cones & Cylinders GCSE revision - Maths4Everyone on TES
• Spheres, Cones & Cylinders, Working Backwards - Maths4Everyone on TES
• Matching graphs and scenarios cards  ( slides ) - Nuffield Foundation
• Conversion graphs - Nuffield Foundation
• Conversion Graphs Practice Grid ( Answers ) - DrAustinMaths
• Plumber's call-out - Nuffield Foundation
• Graphical interpretation - Median Don Steward
• Linear rules, with contexts  - Median Don Steward 
• Bath card sort - adapted from mattsteel87 on TES
• Archimedes' Bath - ColmanWeb
• Desmos Waterline  
• The Language of Functions and Graphs  - Shell Centre
• Kinematics assorted problems - Boss Maths

• 3D geometry - terms and notation - Boss Maths
• Properties of 3D Shapes Practice Strips ( Answers ) - Dr Austin Maths
• Investigating 3D Shapes Worksheet ( Answers ) - Dr Austin Maths
• Faces and elevations activity - Median Don Steward
• Plans and elevations of 3D shapes - Boss Maths
• Plan and Elevation - mathshko
• Drawing in 2D and 3D  (slides/activities) - Dan Walker on TES
• Drawing skills: isometric projection - source unknown, sorry
• Isometric drawing activities - Median Don Steward
• Isometric Drawing - Owen134866 on TES
• Net Tasks - Median Don Steward
• The Box Problem - 1001 Math Problems
• Nets of a Cube - numeracycd.com
• 3D geometry: faces, edges and vertices - Don Steward
• Plotting Coordinates - Ashton Coward
• Coordinate Message  - SMILE
• Coordinates lesson - Boss Maths
• Coordinates - Dan Walker on TES
• Coordinate Practice - Median Don Steward
• Quadrilateral Coordinates - Median Don Steward
• Squares and Coordinates - Median Don Steward
• Graphs - examples and exercises - CIMT
• Coordinates and Shapes - TeachIt Maths
• Coordinate CBSE Questions - Median Don Steward
• Coordinates Problem Solving - White Rose Maths Hub
• Rectangles on a Grid - Don Steward
• Working with sketches - Boss Maths
• Solve geometrical problems on coordinate axes - Boss Maths
• 3D Coordinates examples and exercises - CIMT
• Plotting 3D Coordinates ppt - Teachitmaths
• Lines of symmetry - Median Don Steward
• Reflection symmetry - Boss Maths
• Rotation symmetry - Boss Maths
• Rotational designs  (& interactive activity ) - MathsPad
• Symmetry gifs
• Making symmetrical shapes - Teachit Maths
• Add one square - Median Don Steward
• Symmetry challenge questions ( answers ) - Dr Austin Maths
• Transformations tasks - Dr Austin Maths
• Transformations activities - Teachit Maths
• Transformations lessons - cparkinson3 on TES
• Reflecting diagonally - Median Don Steward
• Reflection lesson - Boss Maths
• Grid reflections - Median Don Steward
• Wordly reflections - Median Don Steward
• Rotations booklet - The Chalk Face
• Rotation questions - Median Don Steward
• Rotations Jigsaw - Tristan Jones on TES
• Rotation lesson - Boss Maths
• Interactive rotation tool and worksheet - MathsPad
• Still pools (reflection) - Median Don Steward
• Vector messages - SMILE
• Translations with vectors - alicescreswick on TES
• Vectors Snakes and Ladders  - MrBartonMaths on TES
• Interactive translation tool and worksheet - MathsPad
• Transformers Activity - danwalker on TES
• Transforming Words - Tristan Jones on TES
• Transformations Challenge - G Westwater on TES
• Transformation Station game - flashymaths.co.uk
• Transformation workbook  - Math4Everyone on TES
• Congruence, similarity and transformations - Boss Maths
• Transformation activities - @DJUdall
• Combinations of transformations - Boss Maths
• Transforming Shapes Codebreaker - Andy Lutwyche
• Invariance activity sheet (new GCSE) - Peter Mattock on TES
• Invariant Points - Miss Konstantine
• Integer enlargement practice
• Fractional enlargement practice
• Negative enlargement practice  
• Enlargement lesson - cparkinson3 on TES
• Positive integer enlargement jigsaw puzzle - Mr Thompson on TES
• Fractional enlargement jigsaw puzzle - Mr Thompson on TES
• Describe the enlargement  - Median Don Steward
• Enlargement lesson - Boss Maths
• Drawing it a bit bigger  - Median Don Steward
• Enlargement Exam Style Questions ( solutions ) - 1st Class Maths
• Negative Enlargement Exam Style Questions  ( solutions ) - 1st Class Maths
• Also see resources listed under 'Similarity' above
• Investigating the sides of right-angled triangles  - Teachit Maths 
• Pythagoras' Theorem - complete topic booklet  - Teachit Maths
• Pythagoras fading examples - evm86 on TES
• Pythagoras practice questions  (includes converse) - Frank Tapson (trol)
• Pythagorean Theorem questions  ( answers ) - @taylorda01
• Pythagoras Puzzle - Dan Walker on TES
• Pythagoras tasks - @giftedHKO
• Pythagoras problems  - The Chalk Face
• Pythagorean Stacks - equationfreak.blogspot.com
• Pythagoras Pile-Up  - @mrandersonmaths 
• Pythagoras' Theorem Student Sheets  (&  notes ) - Nuffield Foundation
• KS3 SAT Pythagoras Questions - Median Don Steward
• Pythagoras interactive activities - MathsPad
• Varied Pythagoras Practice - Nathan Day
• Applied Pythagoras - Dan Draper
• Distance between two points - Corbett Maths
• Pythagorean Triples Tasks  - pythagoreantriples.blogspot.co.uk
• Triple triangle lengths - Median Don Steward
• Pythagoras and surd form - Median Don Steward
• Pythagoraean Theorem in 3D questions  ( answers ) - @taylorda01 
• 3D Pythagoras - Median Don Steward
• 3D Pythagoras Questions - collated from textbooks
• Pythagoras Topic Review Sheet - Maths4Everyone
• Circles and Pythagoras  - National 5 Maths
• GCSE 9-1 Exam Question Practice (Pythagoras) - Maths4Everyone
• Pythagoras exercises - Transum
• Find the hypotenuse problems - Middleton Maths
• Interwoven tasks - Nathan Day
• Star Wars Pythagoras Questions (challenging) - Nathan Day
• Pythagoras Problem Solving Set - see final page for sources
• Pythagoras  Problem Solving Questions  - steele1989
• See my blog post about interesting Pythagoras problems from the new GCSE
• More ideas in my blog post about teaching Pythagoras' Theorem

Right angled

• Trigonometry introduction booklet  - Teachit Maths
• Introduction to Trigonometry  - lesson plans and activities from Project Maths
• Sine, Cosine and Tangent Ratios Fill in the Blanks ( Answers ) - Dr Austin Maths
• Finding Angles Using Trigonometry Fill In The Blanks ( Answers ) - Dr Austin Maths
• Finding Angles Using Trigonometry Name the Film ( Answers ) - Dr Austin Maths
• Finding Lengths Using Trigonometry Fill In The Blanks ( Answers )  - Dr Austin Maths
• Finding Lengths Using Trigonometry Name the Film ( Answers ) - Dr Austin Maths
• Trigonometric ratios questions  - @taylorda01
• Trigonometry Pile Up 1  - Great Maths Teaching Ideas
• Trigonometry booklet  & answers - MissBrookesMaths
• Let's draw some diagrams  - Teachit Maths 
• Trigonometry in Isosceles Triangles  - Mr Thompson on TES
• Finding lengths - faded scaffolding - evm86 on TES 
• SOHCAHTOA slides - Dan Walker
• Right-angled trigonometry - mathshelper.co.uk
• Trigonometry - Median Don Steward
• Trigonometry in isosceles triangles   - Mr Thompson on TES
• Trigonometry worksheets  - Frank Tapson
• Trigonometry tests with answers - Frank Tapson
• Multi-Step Trigonometry Problems Practice Strips ( Answers ) - draustinmaths
• Pirate Trigonometry  - Matthew Kennedy on TES
• MEP trigonometry exercises - CIMT
• Trigonometry and Pythagoras Post It Challenge  - steele1989 on TES
• SOHCAHTOA GCSE revision - Maths4Everyone on TES

Non right angled

• When do we need 1/2absinc? card match  - Teachit Maths
• Trigonometry (area) fill in the gaps - Andy Lutwyche on TES
• Areas of triangles - Cazoom Maths
• Area of a triangle GCSE revision  - Maths4Everyone on TES
• Area of triangle using trig exam style questions ( answers ) - 1st Class Maths
• Sine Rule Introduction  - SMILE
• Sine Rule lesson  - Boss Maths
• Sine Rule - Median Don Steward
• Finding Lengths Using Sine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
• Finding Lengths Using Sine Rule Practice Grid ( Answers ) - Dr Austin Maths
• Finding Angles Using Sine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
• Finding Angles Using Sine Rule Practice Grid ( Answers ) - Dr Austin Maths
• Sine Rule Target Table  - SimplyEffectiveEducation on TES
• Sine Rule exam style questions ( answers ) - 1st Class Maths
• Cosine Rule lesson - Boss Maths
• Cosine Rule - Median Don Steward
• Cosine Rule exam style questions ( answers ) - 1st Class Maths
• Sine Rule Codebreaker - MathsPad (with subscription)
• Sine Rule (ambiguous case) - langy74 on TES
• Finding Lengths Using Cosine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
• Finding Lengths Using Cosine Rule Practice Grid ( Answers ) - Dr Austin Maths
• Finding Angles Using Cosine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
• Cosine Rule Topic Review Sheet  - Maths4Everyone on TES
• Trigonometry collect a joke - Dan Walker on TES
• Trigonometry and bearings  
• Simple Bearings & Trigonometry - Scaffolded worksheet - Mr Thompson on TES
• MEP trigonometry exercises  - CIMT
• Trig Pile Up - MathematicQuinn on TES
• Sine and Cosine rule trigonometry pile up - MrGrayMaths on TES
• Sine and Cosine Rule GCSE questions - Maths4Everyone on TES
• Non-right angled triangles full coverage GCSE questions  - compiled by Dr Frost
• See my blog post New GCSE: Trigonometry Questions for some challenging questions

Graphs and exact values

• Trigonometry worksheets  - Cleave Books (includes trig graph questions)
• Symmetry in Trigonometric Graphs  - pas1001 on TES
• Graphs of trigonometric functions - Boss Maths
• Plotting Trigonometric Graphs Practice Grid  ( Answers ) - Dr Austin
• Trigonometric Graphs Sort It Out ( Answers ) - Dr Austin
• Exact trig values worksheet and codebreaker - MissBrookesMaths
• GCSE Trigonometry - Exact Values - Mr Thompson on TES
• Finding Exact Trig Values - Discovery Learning   - emcnicholl on TES
• Exact Trigonometric Values - Median Don Steward
• Exact Trigonometric Values Crack the Code ( Answers ) - draustinmaths
• Trigonometry without a calculator - joashknight on TES
• Exact Trigonometric Values textbook exercise  - CorbettMaths 
• If you subscribe to MathsPad , they have great exact trig values resources.

3 Dimensional

• 3D Trigonometry lesson - cparkinson3 on TES
• 3D Trigonometry  - Median Don Steward
• 3D Trigonometry - mathshelper.co.uk
• Get 21 - 3D Trigonometry  - HSpedding on TES 
• Trigonometry worksheets  - Cleave Books (includes 3D trig)
• GCSE 9-1 Exam Question Practice (3D Pythagoras + Trigonometry) - Maths4Everyone
• 3D Trig and Pythagoras Exam Style Questions ( Answers ) - 1st Class Maths
• See my blog post on trigonometry for more ideas
• See my post New GCSE: trigonometry questions for some challenging questions
• Upper and lower bounds lesson - Boss Maths
• Error intervals lesson - Boss Maths
• Bounds lesson - Tick Tock Maths
• Upper and lower bounds questions  - The Chalk Face
• Painting bounds  - The Chalk Face
• Highest and lowest bounds - Median Don Steward
• Errors Student Sheet  (&  notes ) - Nuffield Foundation
• Bounds and Error Intervals  - @dooranran on TES
• Lesson plan: bounds and speed - Colin Foster
• Bounds GCSE revision - Maths4Everyone on TES
• Bounds full coverage GCSE questions - compiled by Dr Frost
• Error intervals exercises  - @alcmaths
• Upper and Lower Bounds Fill in the Blanks ( Answers ) - Dr Austin
• More Upper and Lower Bounds Fill in the Blanks ( Answers ) - Dr Austin
• Upper and Lower Bounds Decode the Joke ( Answers ) - Dr Austin
• Upper and Lower Bounds Odd One Out ( Answers ) - Dr Austin
• Upper and Lower Bounds Revision Practice Grid ( Answers ) - Dr Austin
• Upper and lower bounds  - mathshelper.co.uk
• Error Intervals exam style questions ( answers ) - 1st Class Maths
• Upper and Lower Bounds exam style questions ( Answers ) - 1st Class Maths
• See  my blog post about GCSE bounds  for teaching advice
• Vectors resources from Dr Austin Maths
• Adding and subtracting of column vectors - Boss Maths
• Multiplying column vectors by a scalar  - Boss Maths
• Vectors and quadrilaterals - MathsPad
• Vectors problem - The Chalk Face
• Vectors lessons - Dan Walker on TES
• Don Steward's vectors resources
• Vector Hunt -  Peter Mattock on TES
• Defining Vectors - Peter Mattock on TES
• Vector proof - Peter Mattock on TES
• Proofs using vectors - Boss Maths
• Vectors exam questions - bland.in
• Vectors exam questions - teachingmaths.net
• KS3-4 Bridging the gap Pocket 9 - Vectors - AQA All About Maths 
• Harder GCSE vector questions  - Don Steward
• Vectors - Harder GCSE Practice questions for Edexcel 1MA1 syllabus  - THarman on TES
• Vectors workbook - Maths4Everyone on TES
• Vectors exam question practice - Maths4Everyone on TES
• Vectors full coverage GCSE questions  - compiled by Dr Frost
• Vectors Exam Style Questions ( answers ) - 1st Class Maths
• Simple geometric proofs - Boss Maths
• Translating geometric descriptions activity  - topdrawer.aamt.edu.au
• Geometry toolkit  - topdrawer.aamt.edu.au
• Adding auxiliary lines  - topdrawer.aamt.edu.au
• Mathematical Proof (includes geometric proof) & answers - CIMT
• Prove it! (similar triangles) - MathsPad
• Working with sketches  - Boss Maths
• Formula sheet  and sheet showing  formula to learn for GCSE
• Geometry revision from Dr Austin
• Geometry rich venn tasks - mathsvenns.com
• Circle theorem revision cards  -  teachitmaths.co.uk
• Trigonometry revision mat - MathedUp
• Shape and space textbook extract - Oxford University Press
• Exam questions  (various formats, including answers) compiled by Dr Gareth Evans (WJEC)

Volume Calculator

What is volume — volume definition, volume units and conversion table, how to calculate volume — volume formulas, volume calculator and tools dedicated to specific shapes, how do i use the volume calculator, measuring the volume of solids, liquids and gases, how to find the volume of a rectangle vs volume of a box, real-life applications.

The volume calculator will calculate the volume of some of the most common three-dimensional solids. Before we go into how to calculate volume, you must know the definition of volume. Volume differs from the area, which is the amount of space taken up in a two-dimensional figure. So you might be confused as to how to find the volume of a rectangle versus how to find the volume of a box ( spoiler alert: there's no such thing as volume of a rectangle ). The calculator will assist in calculating the volume of a sphere, cylinder, cube, cone, and rectangular solids.

Volume is the amount of space that an object or substance occupies. Generally, the volume of a container is understood as its capacity — not the amount of space the container itself displaces. Cubic meter (m 3 ) is an SI unit for volume.

However, the term volume may also refer to many other things, such as

• the degree of loudness or the intensity of a sound;
• the number or amount of something (usually large quantity); and
• formal word for a book or one in a set of related books.

Popular units of volume are:

• Cubic centimeters (cm 3 )
• Cubic meters (m 3 )
• Liters (l, L)
• Milliliters (ml, mL)
• Fluid ounce (fl oz)
• Cubic inch (cu in)
• Cubic foot (cu ft)
• Quarts (qt)
• Gallons (gal)

If you need to convert the units of volume, you can use our great volume converter . Another useful tool is our grams to cups calculator , which can help if you want to use a food recipe from a different country. Note that it's not a simple conversion but a change from weight (grams) to volume unit (cups) — that's why you need to know the ingredient type (or, more specifically, its density).

Also, you can have a look at this neat volume unit conversion table to find out the conversion factor in a blink of an eye:

There is no simple answer to this question, as it depends on the shape of the object in question. Here are the formulas for some of the most common shapes:

Cube = s 3 s^3 s 3 , where s s s is the length of the side.

Sphere = ( 4 / 3 ) π r 3 (4/3)\pi r^3 ( 4/3 ) π r 3 , where r r r is the radius.

Cylinder = π r 2 h \pi r^2h π r 2 h , where r r r is the radius and h h h is the height.

Cone = ( 1 / 3 ) π r 2 h (1/3)\pi r^2h ( 1/3 ) π r 2 h , where r r r is the radius and h h h is the height.

Rectangular solid (volume of a box) = l w h lwh lw h , where l l l is the length, w w w is the width and h h h is the height (a simple pool may serve as an example of such shape).

Pyramid = ( 1 / 3 ) A h (1/3)Ah ( 1/3 ) A h where A A A is a base area and h h h is the height. For a pyramid with a regular base, another equation may be used as well: Pyramid = ( n / 12 ) h s 2 cot ⁡ ( π / n ) (n/12) h s^2 \cot(\pi/n) ( n /12 ) h s 2 cot ( π / n ) , where n n n is a number of sides s s s of the base for a regular polygon.

Prism = A h Ah A h , where A A A is a base area and h h h is the height. For a right triangular prism, the equation can be easily derived, as well as for a right rectangular prism, which is apparently the same shape as a box.

We've decided to make this volume calculator a simple tool that covers the five most popular 3D shapes. However, not every volume equation and shape type may be implemented here, as it will make the calculator overloaded and unintuitive. So if you're looking for a specific shape, check out the calculators on our site dedicated to volumes of chosen shapes!

Let's look at the example of how to use this volume calculator:

Select the 3D shape type . If you can't find the shape you want to calculate the volume of, choose other special dedicated calculators on our site. In this example, let's assume you want to calculate the volume of a cylinder.

Type your data into the proper boxes . Our cylinder has a radius of 1 ft and a height of 3 ft. You can change the units by a simple click on the unit name.

Here you go! The volume of a chosen shape is displayed . In our case, it's 9.42478 cu ft.

If you want to check how much that is in the US barrels unit, just click on the unit name and choose barrels from the drop-down list. Our cylinder has a capacity of ~2.24 oil barrels.

How to find the volume of objects with different states of matter?

For regular three-dimensional objects, you can easily calculate the volume by taking measurements of its dimensions and applying the appropriate volume equation. If it's an irregular shape, you can try to do the very thing that caused Archimedes to shout the famous word Eureka ! Probably you heard that story — Archimedes was asked to find out if the Hiero's crown is made from pure gold or just gold-plated — but without bending or destroying it. The idea came to him when he was taking a bath — stepping into a bathtub, he noticed that the water level rose. From this observation, he deduced that volume of water displaced must be equal to the volume of the part of his body he had submerged. Knowing the irregular object's volume and its weight, he could calculate the density and compare it with the density of pure gold. Legend says that Archimedes was so excited about this discovery that he popped out of his bathtub and ran naked through the streets of Syracuse.

So, if you want to measure an irregular object's volume, just follow in Archimedes' footsteps (though you can omit the naked race part):

Take a container bigger than the object you want to measure the volume of . It may be a bucket, a measuring cup, a beaker, or a graduated cylinder. It should have a scale.

Pour water into the container and read the volume measurement.

Put the object inside . It should be totally submerged to measure the object's whole volume. Read off the volume. This method won't work if your object dissolves in water.

The difference between the measurements is the volume of our object.

These measurements are essential in calculating the buoyancy force, which are based on Archimedes' principle.

Usually, it's quite easy to measure the volume of a liquid — all you need to have is a graduated measuring vessel of some kind. Choose the one that fits your needs: the amount of liquid and degree of accuracy are the parameters to consider. The containers used in baking a cake will be different from those used in chemistry (e.g., in molar concentration calculations) and will be different from ones used for medical purposes (like the dose of a medication).

We have to use more elaborate methods to measure the volume of a gas. You need to remember that the volume of gas is influenced by temperature and pressure and that gases expand to fill any container in which they're placed. You can try to measure it:

Inflate a balloon with the gas you want to measure (e.g., with helium to lift you up in the air). Then you can use the Archimedes method - put the balloon into the bucket with water and check the volume difference. You'll find detailed instructions on the wikiHow page .

Check out the measures connected to your lung capacity by using a device called spirometer .

In chemistry, a gas syringe is used to insert or withdraw a volume of a gas from a closed system . This laboratory glassware can also be used to measure the volume of gas evolved from a chemical reaction.

Or calculate :

Find the volume of the gas, given its density and mass . Use simple V = m / d V = m / d V = m / d volume equation.

Calculate the volume of a compressed gas in a cylinder by applying the ideal gas equation.

You can't calculate a volume of a rectangle , a volume of a circle, or a volume of a square because they are 2-dimensional geometric figures. As such, a rectangle does not have a volume (but it does have an area). What you're probably looking for is the volume of a rectangular cuboid (or, in more common terms, you want to find the volume of a box), which is a 3-dimensional object.

To find the volume of a box, simply multiply length, width, and height — and you're good to go! For example, if a box is 5×7×2 cm, then the volume of a box is 70 cubic centimeters.

For dimensions that are relatively small whole numbers, calculating volume by hand is easy. For larger or decimal-valued numbers, the use of the volume calculator is very efficient.

There are many applications in real life where the volume calculator is useful:

One such instance is in road or pavement construction , where slabs of concrete must be built. Generally, concrete slabs are rectangular solids, so the concrete calculator — which is an application of the volume calculator — can be used.

Also, the volume formulas may be helpful if you are a keen gardener or just a happy owner of a house with a yard. Check out our other awesome tools, such as potting soil calculator , to estimate the volume and cost of topsoil, as well as the amount of soil needed for your flower pot.

Moreover, you can meet volume in your kitchen or bathroom : any liquid we drink (e.g., bottled water), as well as beauty products or toothpaste, have a volume written on the product packaging (either in milliliters/liters or fluid ounces/gallons).

Next, the volume calculations are really handy if you want to guarantee enough living space for your beloved pets , be they turtles/fish kept in the aquarium or mice/rats in cages.

Another related application, although slightly different, is the concept of surface area . Suppose the entire exterior of a building must be painted. To know how much paint must be purchased, the surface area of the building must be calculated. The convenient to use surface area calculator will calculate this for you.

How to find volume?

The volume formula depends on the shape of the object . One of the most popular shapes is a rectangular prism, also known as a box, where you can simply multiply length times width times height to find its volume. Another common shape is a cylinder — to find its volume, multiply the height of the cylinder by the area of its base (π × r²). For other 3D shapes, check Omni's Volume Calculator.

How do I measure volume?

Measuring the volume depends on your object's state of matter. For liquids , you can use a graduated cylinder or burette for the chemistry lab measurements, or a measuring cup & spoon for everyday life purposes. For gases, to roughly measure the volume, you can inflate a balloon and use it to displace the water in a graduate cylinder. A similar method works for solids — put the object into a graduated container and measure the change in reading.

Is volume squared or cubed?

Volume is "cubed" , as it's a three-dimensional measure. Area is a "squared" value, as the area of a shape covers two dimensions. You can remember that volume is a cubed value by recalling a few volume unit names, such as cubic meters , cubic feet or cubic yards .

How do I calculate volume?

Depending on your object shape, you can use different formulas to calculate volume:

• Cube volume = side³
• Cuboid (rectangular box) volume = length × width × height
• Sphere volume = (4/3) × π × radius³
• Cylinder volume = π × radius² × height
• Cone volume = (1/3) × π × radius² × height
• Pyramid volume = (1/3) × base area × height

What is volume measured in?

A cubic meter is the SI volume unit. However, as it's quite impractical, so most often you will encounter volume expressed in:

• Cubic centimeters
• Cubic inches
• Milliliters

How to find volume of a liquid?

Graduated cylinders and Erlenmeyer flasks will work if you need to roughly measure the volume of a liquid. For more precise measurements, you need to use a volumetric pipette and burette. However, if you're baking a cake or preparing a delicious dish and the recipe uses volume units, you can simply use a measuring cup, glass, or spoon.

What is the SI unit for volume?

The cubic meter (m³) is the SI unit of volume. It is derived from the base SI unit of length — the meter. Although the cubic meter is the basic SI unit, other units are used more often: for the metric system milliliters, liters, or cubic centimeters are popular choices, while for the Imperial system, you can find volume expressed in pints, gallons, cubic inches, cubic feet or cubic yards.

Is volume intensive or extensive?

Volume is an extensive property , the same as the amount of substance, mass, energy, or entropy. An extensive property is a measure that depends on the amount of matter . Check out this example: a glass, a barrel, and a pool full of water have different volumes & masses ( extensive properties ), but water in these three containers will have the same density, refractive index & viscosity ( intensive properties ).

What is the difference between surface area and volume?

A volume is a 3D measure , while surface area is two-dimensional . The volume tells us about the cubic space that an object occupies, and the surface area is the sum of all areas forming the 3D shape. Take the cardboard box as an example 📦:

• Volume is the amount of space taken up by the box — simply, it's the space available inside the box .
• Surface area is the space occupied by the box's sides , calculated when painting the sides or wrapping the box in paper.

How to find the volume of an irregular-shaped object?

You can use the fluid displacement method for irregularly shaped solid objects:

• Fill the container with water and mark the water level.
• Drop your object inside and mark the level again. Make sure your object isn't soluble in water.
• For scaled containers, you can just subtract the original volume from the new volume. And that's it, congrats!

But if your original container doesn't have a scale:

• Take out the object.
• Fill your container with water to the second mark, pour that water into a graduated cylinder/another measuring vessel.
• Repeat step 5 for another marked level and subtract the volumes.
• Pat yourself on the back — you found the volume of an irregular-shaped object!

What does volume measure?

The volume measures the amount of space taken up by an object in three dimensions . Another closely related term is capacity, which is the volume of the interior of the object. In other words, capacity describes how much the container can hold (water, gas, etc.).

What is the volume of the Earth?

The volume of the Earth is approximately equal to 1.08321×10 12 km³ ( 1.08 trillion cubic kilometers ), or 2.59876×10 11 cu mi ( 259 billion cubic miles ). You can get this result using the sphere volume formula (4/3) × π × radius³ and assuming that the Earth's average radius is 6,371 kilometers (3,958.76 mi).

How do I calculate surface area to volume ratio?

To calculate the surface area to volume ratio SA:V, you simply divide the surface area by volume . For some chosen shapes:

• SA:V ratio for cube = (6 × side²) / (side³) = 6 / side
• SA:V ratio for sphere = (4 × π × radius²) / ((4/3) × π × radius³)= 3 / radius
• SA:V ratio for cylinder = (2 × π × radius² + 2 × π × radius × height) / (π × radius² × height) = 2 × (radius + height) / (radius × height)

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