problem solving with 3d shapes

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Here we will learn about 3D shapes, including their names and properties, surface area and volume, and plans and elevations.

There are also 3D shapes worksheets based on Edexcel, AQA and OCR exam questions which are printable, along with further guidance on where to go next if you’re still stuck.

What are 3D shapes?

3D shapes are solid shapes that have 3 dimensions: height, width and depth . 

We can categorise and solve problems involving 3D shapes using a variety of methods.

• 3D shape names

3D shapes can be categorised in several ways.

Firstly a 3D shape is either a polyhedra (a 3D shape with flat polygonal faces), or a non-polyhedra (a 3D shape which has a curved surface). 3D shapes can then be further categorised as shown in the table below. The properties of 3D shapes can help to classify them and recall their names.

Note that some 3D shapes fit into more than one category.

*There are other categories of 3D shapes, but these cover what is needed for the GCSE course.

Here are some examples of common 3D shapes;

Step-by-step guide: 3D shape names

• Faces, edges and vertices

Polyhedra are 3D shapes made up of faces, edges and vertices as illustrated below.

The shape in the table is a cube. It has 6 faces, 12 edges and 8 vertices. 

Step-by-step guide: Faces, edges and vertices

The volume of a 3D shape is the amount of space there is inside the shape.

Step-by-step guide: How to calculate volume

• Surface area

Surface area is the total area occupied by the surface of a 3D shape.

To find the surface area of a prism, cylinder or pyramid it is often helpful to draw the net of the 3D shape, calculate the area of each face, and then add them together.

To find the surface area of 3D shapes with curved surfaces such as spheres and cones (higher GCSE only), there are special formulas you should use

Surface area of a sphere =4 \times \pi \times radius^{2}

Curved surface area of a cone =\pi \times radius \times length

Step-by-step guide: How to calculate surface area

• Plans and elevations

We can use isometric paper to draw 3D shapes.

For example, this 3cm \times 3cm \times 3cm cube has been drawn using the dots as guides.

We can also draw 2D images of 3D shapes from different perspectives. These are called plans and elevations .

A plan is a drawing of what a 3D shape looks like from above looking down (like the bird’s eye view). An elevation is a drawing of what a  3D shape looks like  from the side or front. 

For example,

Draw the plan and side elevation of this 3D shape.

We can also recreate a 3D shape when we are given the plan and elevations . 

Draw the 3D shape which has the following plan and elevations .

Step-by-step guide: Plans and elevations

A prism is a polyhedra that has congruent cross sections . In non-mathematical words, it is a 3D shape that can be sliced like a loaf of bread, and each slice will look like the same flat shape.

For example, a triangular prism has triangular cross sections.

The net of a prism consists of two identical 2D shapes and a number of rectangles.

The number of rectangles is equal to the number of sides in the 2D shape. For example, the net of a triangular prism has 3 rectangles because a triangle has 3 sides. Below are the nets of 3 different prisms.

We can calculate the volume and surface area of prisms.

Step-by-step guide: Prism

A cuboid is a polyhedron with 6 rectangular faces.

Cubes are a special type of cuboid where the length, width and height are all equal.

The net of a cuboid consists of 6 rectangles.

We can calculate the volume and surface area of cuboids.

Step-by-step guide: Cuboid

• Triangular prisms

A triangular prism is a polyhedron consisting of two triangular ends connected by three rectangles. The triangular ends of a triangular prism are congruent (exactly the same).

The net of a triangular prism consists of 3 rectangles and 2 triangles.

We can calculate the volume and surface area of triangular prisms.

Step-by-step guide: Triangular prism

Spheres and hemispheres

A sphere is a 3D shape which can be described simply as a ball. Mathematically it is a non-polyhedra with no vertices and just a single curved surface . Every point on the surface of a sphere is equidistant from the centre. This distance is the radius of the sphere.

A hemisphere is half a sphere. It has one curved surface and one flat circular surface.

We can calculate the volume and surface area of spheres and hemispheres.

Step-by-step guide: Sphere

Step-by-step guide: Hemisphere

Cones are like pyramids but with circular bases. The cross-sections of a cone are similar circles.

We can calculate the volume and surface area of a cone.

Step-by-step guide: Cone

Geometrically a cylinder is like a prism. The cross sections of cylinders are congruent circles.

The net of a cylinder consists of two identical circles and one rectangle that folds around the circles to create a curved surface.

We can calculate the volume and surface area of a cylinder.

Step-by-step guide: Cylinder

A pyramid is a 3D shape which has a flat base shape and an apex.

A pyramid has similar cross-sections as shown in this diagram.

We can calculate the volume and surface area of a pyramid.

Step-by-step guide: Pyramid

See also: Square based pyramid

Volume and surface area of cuboids and prisms

We can work out the volume and surface area of 3D shapes.

The volume of a 3D shape is the space inside it.

The surface area of a 3D shape is the total area of all its faces.

To find the volume of a cuboid we use the formula, 

Calculate the volume of the cuboid.

Length = 6 \ cm

Width = 2 \ cm

Height = 3 \ cm

Volume = 6 \times 2 \times 3 = 36 \ cm^3

The surface area of a cuboid can be calculated by adding together the areas of the six faces. The properties of a cuboid are such that the opposite faces are congruent rectangles.

Therefore we only need to find the area of three different rectangular faces, add them together, and then double this answer to find the total surface area.

Calculate the surface area of the cuboid.

Area front = 6 \times 4 = 24 \ cm^2

Area top = 4 \times 3 = 12 \ cm^2

Area side = 6 \times 3 = 18 \ cm^2

Total surface area 

To calculate the volume of a prism , we calculate the area of the cross-section and then multiply this  by the depth.

Calculate the volume of the prism.

This cross-section is a triangle.

Area triangle = \frac{1}{2}(\text{base} \times \text{height})

Area of the cross-section,

The surface area of a prism can be calculated by adding together the areas of all its faces.

Calculate the surface area of the prism.

This is a triangular prism. It has 5 faces.

Area of each triangular face = \frac{1}{2} (4 \times 3) = 6 \ cm^2

Area of base rectangle = 6 \times 4 = 24 \ cm^2

Area of vertical side rectangle = 3 \times 6 = 18 \ cm^2

Area of sloping side rectangle = 6 \times 5 = 30 \ cm^2

Total surface area = 6 + 6 + 24 + 18 + 30 = 84 \ cm^2

Volume of cylinders, pyramids, cones and spheres

Geometrically a cylinder is like a prism, so we can calculate the volume using the same formula.

The cross-section of a cylinder is a circle and so we need to remember the formula for area.

For example, calculate the volume of the cylinder .

(Note that ‘height’ and ‘depth’ can be used interchangeably here.)

We can calculate the volume of a pyramid using this formula.

For example, calculate the volume of the pyramid .

Area of the base =6 \times 6 = 36 \ cm^2

We can calculate the volume of a cone using this formula.

For example, calculate the volume of the cone .

We can calculate the volume of a sphere using the formula.

Volume = \frac{4}{3} \pi r^{3}

For example, calculate the volume of the sphere .

How to use 3D shapes

3D shapes is a broad topic which covers many different facts and skills.

We will learn about:

• Hemispheres

Explain how to use 3D Shapes

3D shapes worksheet

Get your free 3D shapes worksheet of 20+ questions and answers. Includes reasoning and applied questions.

3D shapes examples

Example 1: 3d shape names.

What is the name of this 3D shape?

• Identify if the 3D shape is, i)  a polyhedron (all flat polygonal faces) – go to step 2. ii) a non-polyhedron (includes a curved surface) – go to step 3.

This 3D shape has a curved surface; go to step 3. 

2 Identify if all the faces are the same regular shape. i) If yes, this is one of the platonic solids (tetrahedron, cube, octahedron, dodecahedron or icosahedron). ii) If no, continue to step 3.

Skip this step.

3 Identify if the 3D shape is a) a pyramid or cone (3D shape with a base, an apex and similar cross-sectional areas). b) a prism or cylinder (3D shape with congruent cross-sectional areas). c) neither.

This 3D shape has a base, an apex and similar cross sectional areas; it is a pyramid.

4 Name the shape. a) For pyramids: the name of the base shape often forms the name of the 3D shape. b) For prisms: the name of the cross-sectional area often forms the name of the 3D shape. c) i) For neither (polyhedron): this is an irregular polyhedron or a compound solid. ii) For neither (non-polyhedron): those needed to be named on the GCSE syllabus are spheres and hemispheres .

Answer: Cone

Example 2: faces, edges and vertices

How many faces, vertices and edges does this 3D shape have?

Inspect the shape to visualise its faces / edges / vertices.

This 3D shape is a prism. Below is a sketch of the net.

Count the number of faces / edges / vertices.

This 3D shape has 6 faces.

There are 4 edges going around the front polygon, there are 4 edges going around the back polygon, there are 4 edges joining the front and back polygons.

This gives a total of 12 edges.

There are 4 vertices at the bottom of the 3D shape. There are 4 vertices at the top of the 3D shape. This gives a total of 8 vertices.

Answer: 6 faces, 12 edges, 8 vertices.

Example 3: volume of a sphere

Calculate the volume of a sphere with radius r=4.5cm. Write your answer correct to 2 decimal places.

Write down the formula for the volume of a sphere.

The volume of a sphere

V=\frac{4}{3}\pi{r^{3}}.

Substitute the given values into the formula.

Substituting r=4.5cm, into V, we have

V=\frac{4}{3}\times\pi\times{4.5^{3}}.

Complete the calculation.

V=381.70351

Write the answer, including the units.

The question asks for the answer to be correct to two decimal places. 

The unit of length in the question is cm, so the answer for the volume will be in cm^{3}.

V = 381.70cm^{3}

Example 4: plans and elevations

On square paper draw the plan, front elevation and side elevation of this 3D shape.

Draw the plan view.

The plan view is taken looking down onto the shape. We should only be able to see blue squares.

The plan view consists of 3 blue squares horizontally in line with each other.

Draw the front elevation.

The front elevation is taken looking from the left of the shape (as labelled on the diagram). We should only be able to see pink squares.

The front elevation consists of the three rows of 1, 2, and 3 pink squares in line with each other.

Draw the side elevation.

The side elevation is taken looking from the right of the shape. We should only be able to see green squares.

The side elevation consists of 3 green squares vertically in line with each other.

Example 5: surface area

Calculate the total surface area of the cylinder below. Write your answer in terms of \pi.

Work out the area of each face.

The net of the cylinder consists of two circles and a rectangle where the width of the rectangle is the circumference of the circle, and the height of the rectangle is the height of the prism.

The circumference of the circle is C=2\pi{r}=2\times\pi\times{3}=6\pi .

Sketching and labelling the net of the cylinder, we have

The area of the top and bottom of the cylinder are the same, moreover, they are two circles with a radius of 3cm . Using the formula for the area of a circle ( A=\pi{r}^{2} ) where r = 3 , we have

A=\pi\times{3}^{2}=\pi\times{9}=9\pi .

As there are two circles, 9\pi\times{2}=18\pi .

Important note: We are leaving the working in terms of π as the answer is also in terms of π .

The remaining rectangular face has an area of

\text{area of a rectangle }=\text{height }\times\text{width }=5\times6\pi=30\pi .

Add the area of each face together.

The units of length in the question are in centimetres and so the units of area in the solution are in square centimetres ( cm^{2} ). 

The surface area of the cylinder in terms of π is 48 \pi cm^{2} .

Example 6: volume

Calculate the volume of the triangular prism below.

Write down the formula.

The volume of a triangular prism is

V=\text{area of the triangular cross section }\times\text{length}.

Calculate the area of the triangular cross section and substitute the values.

The area of a triangle is A=\frac{b\times{h}}{2} .

As b=10cm and h=13cm , substituting these into A we get

A=\frac{10\times{13}}{2}=65 .

We now have A=65cm^{2} and l=16cm which we can substitute into the formula for the volume,

V=\text{area of the triangular cross section}\times\text{length}=65\times{16} .

Work out the calculation.

The units in the question are in centimetres and so the solution is in cubic centimetres ( cm^{3} ).

Common misconceptions

• Using a descriptive 3D shape name instead of the correct mathematical name

Some 3D shapes have descriptive names or can be likened to certain categories of 3D shapes but must be referred to by the correct mathematical name. Here is a list of examples which are part of the GCSE syllabus.

Descriptive name/Incorrect name = correct mathematical name

Circular prism = cylinder

Rectangular prism = cuboid

Circular based pyramid = cone

Triangular based pyramid (equilateral triangles) = tetrahedron

Ball = sphere

Half a ball = hemisphere

• Using the wrong formula

There are several formulas that are given on the formula sheet for surface area and volume. Make sure you select the right one by correctly identifying the 3D shape in the question and what it is asking you to calculate.

• Using a mixture of units in a calculation

When you are calculating volume or surface area you must ensure that the units of length you are using are the same. This may mean that you need to convert a unit before you carry out a calculation. For example, if one length is given in metres and another is given in centimetres, you must convert one of them so that they are either both in metres or both in centimetres.

• Missing or incorrect units

You should always include units in your final answer.  Surface area is measured in units squared (e.g. mm^{2}, \ cm^{2}, \ m^{2} etc). Volume is measured in units cubed (e.g. mm^{3}, \ cm^{3}, \ m^{3} etc). For volume you may also be asked to give your answer in litres or millilitres for example. Note that 1 \ ml= 1 \ cm^{3} 1 \ litre = 1000 \ ml = 1000 \ cm^{3}

• Hidden faces, edges and vertices

When counting the faces, edges or vertices on an image of a 3D shape you should consider carefully if there are any that are, in a sense, ‘hidden’. Some 3D drawings use dashed lines to represent edges that cannot be seen but sometimes they do not. In this case try and visualise if there are any ‘hidden’ faces, edges or vertices behind or underneath the 3D representation.  

• Rounding at multiple stages during long calculations

It is important not to round decimal answers until the end of a calculation. Rounding at multiple stages will result in an inaccurate final answer. It is a good idea to learn how to use the ‘ANS’ button or memory function on your calculator which will help you to substitute long decimal answers into subsequent calculations. 

• Mixing up the front and side elevations

The front elevation and the side elevation can be labelled in different ways. The front elevation is often indicated by an arrow, like in the picture below. Always look for an indication like this and do not make your own assumption about which perspective is the front.

Practice 3D shapes questions

1. What is the name of this 3D shape?

Pentagonal Pyramid

Pentagonal Prism

Irregular Prism

This 3D shape has congruent cross-sectional areas, it is therefore a prism. The shape of the cross-section is a regular pentagon, hence, it is a pentagonal prism.

2. This is a square based pyramid. How many vertices does it have?

The base of the pyramid has 4 vertices as they are the corners of a square. The apex (the highest point if the shape) is another vertex, totaling 5.

3. Calculate the volume of a hemisphere with radius 5cm. Write your answer in terms of \pi .

The volume of a sphere is V=\frac{4}{3}\pi r^{3}.

A hemisphere is half this value.

4. Which of the following represents the plan view of the 3D shape below:

The plan view is from above looking down, the bird’s eye view.

5. Calculate the surface area of the triangular prism below.

Find the area of each face and then add these values together.

6. Calculate the volume of the cuboid below.

The volume of a cuboid is equal to V = h \times w \times d.

Here, V = 3.8 \times 9.2 \times 4.3 = 150.328 \ cm^{3} (remember cube units for volume).

3D shapes GCSE questions

1. Match the solid shape to the correct mathematical name.

For every two correct matches.

2. (a) Determine which shape has the greater volume?

(b) State the volume to surface area ratio of Shape A in its simplest form.

Volume of Shape A = 4^{3} = 64 \ cm^{3}

Volume of Shape B = \pi \times r^{2} \times h = \pi \times 2.6^{2} \times 3 = 63.7 \ cm^{3}

Shape A has a greater volume.

3. The sun is almost spherical. The diameter of the sun is approximately equal to 1.4\times 10^{6} \ km.

(a) Calculate the radius of the sun in standard form.

(b) Given that the volume of a sphere is equal to V=\frac{4}{3}\pi r^{3}, estimate the volume of the sun.

Write your answer in standard form correct to 2 significant figures.

  V=\frac{4}{3}\times\pi\times(7\times{10^{5}})^3

Learning checklist

You have now learned how to:

• Identify 3D shapes, including cubes and other cuboids, from 2D representations
• Construct and interpret plans and elevations of 3D shapes
• Calculate surface areas and volumes of spheres, pyramids, cones and composite solids
• Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3D
• Recognise when it is possible to use formulae for area and volume of shapes
• Compare and classify geometric shapes based on their properties and sizes

The next lessons are

• Pythagoras theorem
• Trigonometry
• Circle theorems

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3D Geometry

Entering the 3rd dimension!

Cuts Through Shapes

Think about 3D shapes by cutting them into pieces.

Surfaces of Shapes

Fold and fold again to transform 2D shapes into 3D.

Pieces of 3D

Slice, extrude and transform 3D shapes into different configurations.

End of Unit 1

Complete all lessons above to reach this milestone.

0 of 3 lessons complete

Introduction to Nets

Fold up nets to make 2D shapes into 3D. Unfold them to see how the faces relate.

Nets of a Cube

What nets can successfully fold up to make a cube?

Exploring Cubes

Explore the faces of a cube and use nets to see how they relate.

• Platonic Solids

Discover how many of these symmetrical solids can be constructed.

Lines Through Cubes

Apply the Pythagorean theorem to 3D distances.

3D Shortest Distance

How can the shortest distance on the surface of a 3D shape be found?

Strings and Ants

Puzzle out these 3D distance problems by unfolding the shapes.

End of Unit 2

0 of 7 lessons complete

Introduction to Cross Sections

Think like an MRI machine as you slice through these shapes.

Building Intuition

How do cross sections relate to the shape they come from?

Cross Sections of Cubes

Explore the variety of shapes that can be obtained just by slicing up a cube.

Predicting Solids

Can the cross sections of a solid reveal its full shape?

Halves of Solids

How many ways are there to cut a 3D solid in half?

Other Fractions of Cubes

Stretch and test your understanding with these cube fraction puzzles.

End of Unit 3

0 of 6 lessons complete

Vertices, Edges, and Faces

Is there a pattern here?

Uniform Vertex Configurations

Examine polyhedra that have the same polygons in the same order at every vertex.

Cutting Solids

Keep cutting solids and see what happens to the shapes as they transform.

• Euler's Formula

Discover the formula that describes the relationship between faces, edges, and vertices.

Proving Euler's Formula

See why Euler's formula must always be true.

Explore the connections between dual polyhedra and the ways they relate.

End of Unit 4

Course description.

Explore the fundamental concepts of three-dimensional geometry: What strangely-shaped 3D pieces can result from slicing up 3D polyhedra with planes? What flat polygons can fold up into 3D shapes? If you're running around on the surface of a cube-world, what's the shortest path between two opposite corners? (The answer to this last one might surprise you.) In this course, you'll stretch problem-solving techniques from flat figures into a third-dimension and explore some mathematical ideas and techniques completely unique to 3D geometry. For example, you'll investigate and learn how to apply Euler's facet counting formula, a formula which describes a surprising algebraic relationship that relates the number of corners, edges, and faces that any polyhedron can have. To succeed at this course, you should already have some familiarity with the basics of 2D geometry. Additionally, some algebra is used in this course, but nothing beyond the level of Algebra I.

Topics covered

• Cross Sections
• Dissecting Shapes
• Distance in 3D
• Dual Polyhedra
• Paths on a Surface

Prerequisites

• Geometry II
• Beautiful Geometry

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3d Shapes Activities

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3D Shapes Volume Problems

Problems on 3D shapes, such as prisms, cube, cylinder, volume are presented along with detailed solutions

• 5th Grade Math
• Solid Figures

Mastering 3D shapes with Grade 5 solid figures worksheets

Do you want to boost your 5 th grader’s confidence in mastering 3D shapes ? Then this article is for you. With Mathskills4kids’ Grade 5 solid figures worksheets practice, your students learn how to identify and classify three-dimensional shapes using their properties , draw and model three-dimensional shapes using nets and solids , and calculate the surface area and volume of three-dimensional shapes in no time.

Do you also wish for your students to learn how to apply this vital concept of 3D geometry to real-world problems? Stay engaged in this article and enjoy our fun strategies. But before then, let’s find out what three-dimensional shapes are and why they are important.

Enhance geometry learning with our Grade 5 three-dimensional worksheets for homeschooling and classroom use.

Hello math lovers! Today, we are excited to offer you this remarkable collection of Grade 5 three-dimensional worksheets for homeschooling and classroom use to enhance geometry learning skills in 5 th graders.

Geometry is one of the most fascinating and practical branches of mathematics. It helps us understand the shapes and structures of the world around us and how to measure and manipulate them.

In grade 5, students are introduced to three-dimensional shapes, which have length, width, and height. These three-dimensional shapes are also called solids or polyhedral.

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• Number sense
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What are three-dimensional shapes, and why are they important.

Three-dimensional shapes are shapes that have length, width, and height. They can be thought of as containers that can hold objects or liquids. Some examples of three-dimensional shapes are cubes, spheres, cones, cylinders, pyramids, prisms, and polyhedrons.

Three-dimensional shapes are important because they help us model and understand the physical world. For example, we can use cubes to build houses, spheres to represent planets, cones to make ice cream, cylinders to store water, pyramids to design monuments, prisms to refract light, and polyhedrons to create sculptures.

Three-dimensional shapes also help us develop spatial reasoning skills, perfect for solving problems in science, engineering, art, and everyday life.

How to identify and classify three-dimensional shapes using their properties

One way to identify and classify three-dimensional shapes is by using their properties . Properties are characteristics or features that describe a shape. Some typical properties of three-dimensional shapes are:

• Faces : The flat surfaces that make up a shape. For example, a cube has six faces that are all squares.
• Edges : The line segments where two faces meet. For example, a cube has 12 edges equal in length.
• Vertices : The points where three or more edges meet. For example, a cube has eight vertices that are all right angles.
• Base : The face or faces a shape rests on or is built from. For example, a cone has one base that is a circle.
• Apex : The vertex or vertices opposite to the base or bases. For example, a cone has one apex that is a point.
• Cross-section : The shape formed when a plane cuts through a shape. For example, the cross-section of a cylinder can be a circle or an ellipse.

We can compare and contrast different three-dimensional shapes and group them into categories using these properties. For example, we can say that a cube and a rectangular prism belong to the same category because they both have six rectangular faces, 12 edges, and eight vertices.

However, they are different because a cube has all equal faces, edges, and angles, while a rectangular prism does not.

How to draw and model three-dimensional shapes using nets and solids

Another way to identify and classify three-dimensional shapes is using nets and solids. Nets are two-dimensional patterns that can be folded or cut out to form three-dimensional shapes. Solids are three-dimensional models made from materials like clay, paper, cardboard, or plastic.

Nets and solids help us visualize and represent three-dimensional shapes in different ways. For example, we can use nets to show how the faces of a shape are connected and arranged. We can also use solids to show the shape's appearance in real life.

To draw a net of a three-dimensional shape , we need to follow these steps:

• Identify the faces of the shape and their dimensions.
• Draw each face as a polygon on a flat surface.
• Connect the faces with tabs or flaps along their edges.
• Label the faces with letters or numbers for reference.

To make a solid from a net of a three-dimensional shape, we need to follow these steps:

• Cut out the net along its outline.
• Fold the net along its edges.
• Glue or tape the tabs or flaps together.
• Check if the solid matches the shape.

How to teach 3D shapes to 5th graders?

There are many ways to teach 3D shapes to 5th graders , but here are some of the most effective and fun ones:

You can ask students to identify the shapes, count their faces, edges, and vertices, compare their sizes and volumes, build new shapes from them, etc.

You can also ask students to create drawings and diagrams of 3D shapes using rulers, compasses, protractors, etc.

• Use games and puzzles : Games and puzzles are interactive activities that make learning fun and challenging. They help students practice their skills and apply their knowledge in different situations. You can use games and puzzles to teach 3D shapes in various ways, such as:
• Matching games : Students have to match 3D shapes with their names or properties
• Sorting games : Students have to sort 3D shapes into different categories or groups based on their attributes
• Memory games : Students have to remember the location or features of 3D shapes on a grid or a board
• Tangram puzzles : Students have to use 2D shapes to form 3D shapes
• Jigsaw puzzles : Students have to assemble pieces of a net or a picture of a 3D shape
• Logic puzzles : Students have to use clues or rules to find out the missing information about a 3D shape

How to calculate the surface area and volume of three-dimensional shapes

Another vital skill students need to learn in grade 5 is calculating three-dimensional shapes' surface area and volume . The surface area is the total area of all the faces of a shape, and the volume is the amount of space that a shape occupies. These measurements can help us compare, design, and build different objects using three-dimensional shapes.

To calculate the surface area and volume of three-dimensional shapes , we need to use some formulas that depend on the shape and its dimensions. For example;

• The surface area of a cube is 6s 2 , where s is the length of a side, and the volume of a cube is s 3 .
• The surface area of a rectangular prism is 2(lw + lh + wh), where l, w, and h are the prism's length, width, and height, and the volume of a rectangular prism is lwh.
• The surface area of a sphere is 4πr 2 , where r is the radius of the sphere, and the volume of a sphere is (4/3)πr 3 .

To find the surface area and volume of more complex shapes , such as pyramids, cones, and cylinders, we can use strategies such as decomposing them into simpler shapes, using nets or formulas involving other measurements such as the base area or the slant height. For example;

• The surface area of a pyramid is B + (1/2)pl, where B is the base area, p is the perimeter of the base, and l is the slant height
• The volume of a pyramid is (1/3)Bh, where h is the pyramid's height.

How to apply the concepts of three-dimensional geometry to real-world problems

Learning how to work with three-dimensional shapes is fun and useful for solving real-world problems. For instance, we can design and build structures such as houses, bridges, towers, or sculptures using three-dimensional geometry.

We can also use three-dimensional geometry to measure and compare objects like boxes, cans, balls, or bottles. We can also use three-dimensional geometry to explore and understand natural phenomena such as crystals, planets, stars, or snowflakes.

Some examples of real-world problems that involve three-dimensional geometry are:

• How much paint do we need to cover a cube-shaped room with a side length of 10 feet?
• How many tennis balls can fit inside a cylindrical container with a radius of 6 inches and a height of 12 inches?
• What is the volume of cone-shaped ice cream with a radius of 2 inches and a height of 4 inches?
• How many faces does a dodecahedron have? How many vertices? How many edges?
• What is the shape of a snowflake? How can we draw it using a net?

Bonus: where to find additional resources to reinforce 5 th graders’ 3D shapes skills

If you want to find more resources to help your 5th graders learn about 3D shapes, here are some websites that you can check out:

• Math is Fun ( https://www.mathsisfun.com/geometry/solid-geometry.html ): This website has a lot of information and examples about 3D shapes, as well as interactive quizzes and games.
• Khan Academy ( https://www.khanacademy.org/math/basic-geo/basic-geo-volume-sa ): This website has video lessons and exercises about the surface area and volume of 3D shapes.
• IXL ( https://www.ixl.com/math/grade-5/properties-of-three-dimensional-figures ): This website has practice questions and feedback about the properties of 3D shapes.
• Math Salamanders ( https://www.math-salamanders.com/3d-geometric-shapes.html ): This website has printable worksheets and nets of 3D shapes.

Thank you for sharing the links of MathSkills4Kids.com with your loved ones. Your choice is greatly appreciated.

In this article, we have learned about three-dimensional shapes and how to enhance mastering 3D Shapes learning with our Grade 5 solid figures worksheets . We have learned how to identify and classify three-dimensional shapes using their properties, draw and model them using nets and solids, calculate their surface area and volume, and apply their concepts to real-world problems.

We hope that you have enjoyed this article and that you will try Grade 5 Mathskills4kids’ solid figures worksheets with your students or children. They are fun, engaging, and educative!

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Resources tagged with: Nets

There are 23 NRICH Mathematical resources connected to Nets , you may find related items under 3D geometry, shape and space .

The Spider and the Fly

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

A Puzzling Cube

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

Tet-trouble

Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

Which Face?

Which faces are opposite each other when this net is folded into a cube?

Cool as Ice

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

Can Jo make a gym bag for her trainers from the piece of fabric she has?

Make Your Own Pencil Case

What shape would fit your pens and pencils best? How can you make it?

Chopped Dice

Can you make a new type of fair die with 14 faces by shaving the corners off a cube?

Auditorium Steps

What is the shape of wrapping paper that you would need to completely wrap this model?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Triangular Faces

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

All Wrapped Up

What is the largest cuboid you can wrap in an A3 sheet of paper?

All Is Number

Read all about Pythagoras' mathematical discoveries in this article written for students.

Thinking 3D

How can we as teachers begin to introduce 3D ideas to young children? Where do they start? How can we lay the foundations for a later enthusiasm for working in three dimensions?

More Christmas Boxes

What size square should you cut out of each corner of a 10 x 10 grid to make the box that would hold the greatest number of cubes?

Cubic Conundrum

Which of the following cubes can be made from these nets?

Cutting a Cube

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

Rhombicubocts

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

Let's Face It

In this problem you have to place four by four magic squares on the faces of a cube so that along each edge of the cube the numbers match.

Face Painting

You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.

Platonic Planet

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?