volume of cube and cuboid problem solving

Volume of cubes and cuboids

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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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Problem Solving with Cuboids

October 14, 2022.

In this lesson, five problems link the volume of cuboids to:

• 3D coordinates
• Standard form
• Setting up and solving equations
• Converting between metric units

Throughout the lesson, I asked students to sketch the diagrams so they could label the critical information. Then, I encourage them to work together to share their approaches to each problem and question each other. While most calculations are reasonably simple, I remind students to use a calculator to focus on problem-solving rather than arithmetic.

I recommend teaching this lesson as part of the schemes of work on  perimeter, area and volume  in key stage 3 or  volume and surface area  in Key Stage 4.

Problem Solving with Cuboids and 3D Coordinates

Finding a length when given the volume and area is a common question when learning about cuboids. This question takes the idea up another level by challenging students to find the cross-section area from two 3D coordinates and use the calculated length to find the third coordinate.

If students struggle to get started, I encourage them to sketch the cuboid without the grid and note the cross-sectional area and length FE.

Next, I prompt them to label coordinates C and F on the diagram to find the area of face BFGC.

Problem Solving with Cuboids and Ratio

I’ve yet to find a topic that can not be linked to ratio in sIn this question, students consider which two parts of a three-part ratio combine to give the smallest area.

Most students correctly identify the numbers 2 : 3 but often struggle knowing how to use them to give the area of 150 cm2. If this happens, I encourage them to think about equivalent ratios, for instance, 4 : 6 or 6 : 9 and so on.

Problem Solving with Cuboids and Unknowns

In my experience, because there is limited information, students find this question the most challenging. Therefore, I encourage them to begin by finding the length of the pink cube.

Next, they need to consider what changes and remains the same without the blue cuboid. When students realise both shapes have the same height, they can find the blue cross-sectional area. From this, go on to work out x.

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About Mr Mathematics

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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Volume Of A Cuboid

Here we will learn about the volume of a cuboid, including how to calculate the volume of a cuboid and how to find missing lengths of a cuboid given its volume.

There are also volume and surface area of a cuboid worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is volume of a cuboid?

The volume of a cuboid is the amount of space there is within a cuboid. 

Cuboids are 3 dimensional shapes with 6 rectangular faces.

The formula for the volume of a cuboid is:

Volume is measured in cubic units, e.g. mm^3, cm^3 or m^3.

This cuboid is made from 24 unit cubes.

Its volume is

The units of volume are cubed units

E.g. mm^3 (cubic millimetres), cm^3 (cubic centimetres), m^3 (cubic metres) .

How to calculate the volume of a cuboid

In order to calculate the volume of a cuboid:

• Write down the formula. Volume = length \times width \times height

2 Substitute the values into the formula.

Make sure the units are the same for all measurements

3 Substitute the values into the formula.

4 Write the answer, include the units.

Volume of a cuboid worksheet

Get your free volume of a cuboid worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Volume of a cuboid examples

Example 1: volume of a cuboid.

Work out the volume of the cuboid:

• Write down the formula.

Here the length is 10 \; cm , the width is 2 \; cm and the height is 5 \; cm .

3 Work out the calculation.

The measurements are in cm therefore the volume will be in cm^3.

Example 2: volume of a cube

Work out the volume of this cube:

Since this is a cube, the length of the cuboid, width of the cuboid and height of the cuboid are all 6 \; cm.

Volume = 6 \times 6 \times 6

Example 3: volume of a cuboid (different units)

Work out the volume of this cuboid:

Notice here that one of the units is in cm whilst the others are in m. We need all the units to be the same to calculate the volume. 

We can change cm to m: 50cm = 0.5m.

Now that we have all of the measurements in m, we can calculate the volume:

Volume = 4 \times 2 \times 0.5

Since the measurements that we used were in metres, the volume will be in cubic metres.

Volume=4 \; m^3

How to work out a missing length given the volume

Sometimes we might know the volume and some of the measurements of a cuboid and we might want to work out the other measurements. We can do this by substituting the values that we know into the formula for the volume of a cuboid and solving the equation that is formed.

2 Substitute the values that you do know into the formula.

3 Solve the equation.

Missing length given the volume examples

Example 4: find the width of a cuboid given the volume.

The volume of the cuboid is 56 \; cm^3 . Work out the width of the cuboid.

In this example, we know that the volume of the cuboid is 56 cubic centimetres, the width is 7 \; cm and the height is 2 \; cm.

Substituting these into the formula:

\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ 56&=7 \times w \times 2\\\\ 56&=14w \end{aligned}

Since the measurements in this question were in cm and cm^3 , the width will be in cm

Example 5: find the length of a cuboid given the volume

The cuboid below has a square base. The height of the cuboid is 8 \; m and the volume of the cuboid is 32 \; m^3 . Find the length of the cuboid.

In this example, we know that the volume of the cuboid is 32 cubic metres and the height is 8 \; cm .

\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ 32&=l \times w \times 8\\\\ 32&=8lw \end{aligned}

Since the base is a square, we know that the length and width are the same.

Therefore we are looking for a number that, when multiplied by itself, makes 4. We need to find the square root of 4.

The length and width of this cuboid are 2.

Example 6: dimensions of a cube given the volume

Work out the dimensions of a cube which has a volume of 64 \; cm^3

The only value we currently know is the volume is 64 cubic cm.

\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ 64&=l \times w \times h \end{aligned}

Since the shape is a cube, we know that the length, width and height are all the same. Therefore we are looking for a number that, when multiplied by itself three times, makes 64. We need to find the cube root of 64.

\sqrt[3]{64}=4

The length, width and height of this cube are 4.

The cube is 4cm \times 4cm \times 4cm

Common misconceptions

• Missing/incorrect units

You should always include units in your answer. Volume is measured in units cubed (e.g. mm^3, cm^3, m^3 etc)

• Calculating with different units

You need to make sure all measurements are in the same units before calculating volume. E.g. you can’t have some in cm and some in m

• Dividing by three rather than cube rooting

If you are given the volume of a cube and you need to find the side length, remember the inverse of cubed is cube root, not divide by 3.

E.g. if the volume of a cube is 8cm^3 , the side length is \sqrt[3]{8}=2 \; \mathrm{cm} (not 8\div3 )

Related lessons

Volume of a cuboid is part of our series of lessons to support revision on cuboid. You may find it helpful to start with the main cuboid lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Surface area of a cuboid
• Volume of a cube
• Surface area of a cube

Practice volume of a cuboid questions

1. Work out the volume of the cuboid

2. Work out the volume of the cube

3. Work out the volume of this cuboid

Here there are measurements in cm and m so we must make the units the same before calculating the volume.

We can convert 380 \; cm to 3.8 \;m.

Since the measurements we used were in metres, the volume will be in cubic metres.

4. The volume of this cuboid is 600 \; cm^{3} .

Work out the height of the cuboid.

5. The base of this cuboid is a square. The volume of the cuboid is 450\; cm^{3} .

Since the base of the cuboid is a square, the length and the width are both 10 \; cm.

6. The volume of this cube is 343 \; cm^{3} .

What is the length of the cube?

Since this is a cube we know that the length, width and height are all equal. Therefore we are looking for a number that, when multiplied by itself 3 times, makes 343. We need the cube root of 343.

The length, width and height of the cube are all 7 \;cm.

Volume of a cuboid GCSE questions

1. Work out the volume of this cuboid. Give your answer in cm^3.

2. A paddling pool is in the shape of a cuboid.

(a) Work out the volume of the paddling pool.

(b) Sam wants to fill the pool so that it is \frac{5}{6} full.

Water flows out of her hose pipe and into the pool at a rate of 20 litres per minute.

Given that 1\;\mathrm{m}^{3}=1000 \;\mathrm{l} , calculate the length of time it would take Sam to fill the pool so that it is \frac{5}{6} full. Give your answer in hours.

(a) \text{Volume }=4 \times 1.8 \times 0.6

3. A carton of orange juice is shown below. The carton is completely full. The orange juice is poured into another container, as shown below.

What will the height of the orange juice in the container be?

Learning checklist

You have now learned how to:

The next lessons are

• Volume of a prism
• Pythagoras’ theorem
• Volume of a triangular prism

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Volume of Cubes and Cuboids

In volume of cubes and cuboids we will discuss how to calculate volume in different questions.

What is a Volume?

The volume of any 3-dimensional solid figure is the measure of space occupied by the solid. In case of a hollow 3-dimensional figure, the volume of the body is the difference in space occupied by the body and amount of space inside the body.  We also come across different hollow objects in our daily life. These hollow objects can be filled with air or liquid that takes the shape of the container. Here, the volume of the air or the liquid that the interior of the hollow object can accommodate is called the capacity of the hollow object. 

Thus, the measure of space an object occupies is called its volume. The capacity of an object is the volume of substance its interior can accommodate. 

●  The units for measuring volume are  cubic units , i.e., cm², m², etc.  ●  The volume can be measured in  litres  or  milliliters . In such cases, volume is known as capacity. 

Standard Unit of Volume: Volume is always measured in cubic units. The standard unit volume is 1 cm³ but there are various other units of measurement of length like m, dm, dam, etc., so we have many other standards of measurement of volume.

Let’s observe the chart to understand the relation between the various units of volume.

A cuboid is made of six rectangular regions called faces. It has 6 faces. They are ABCD (top face), EFGH (bottom face), ABGH (front face), DEFC (back face), ADEH and BCFG are side faces. 

Thus, a cuboid is made up of 3 pairs of congruent rectangular faces (top, bottom); (front, back); (side face). 

Face EFGH is called the base of the cuboid.  Front face ABGH, back face DEFC and side faces ADEH and BCFG are called the lateral faces of the cuboid. 

Any two faces other than opposite faces meet in a line segment which is called an edge of the cuboid. The cuboid has 12 edges AB, BC, CD, DA, EF, FG, GH, HE, AH, BG, DE and CF. The three edges meet at a common point called the vertex. A cuboid has 8 vertices, namely A, B, C, D, E, F, G and H. 

Now we will discuss about the volume of cubes and cuboids.

Volume of cuboid: 

Let l, b, h represent length, breadth and height of the cuboid.

Area of the rectangular base EFGH of the cuboid = l × b.

Volume of the cuboid = (Area of base) × (height of the cuboid) = (l × b) × h = lbh

Let us consider a cuboid of length ‘l’, breadth ‘b’ and height ‘h’.

Then the volume of the cuboid is given by ………… ● Volume = length × breadth × height

● Length of cuboid = Volume/(breadth × height)

● Breadth of the cuboid = Volume/(length × height)

● Height of the cuboid = Volume/(length × breadth)

While finding the volume of cuboid, length, breadth and height must be expressed in the same units.

Volume of cube: It is a special type of cuboid whose length, breadth and height are equal. So, the volume of the cube whose edge is l is expressed as ……….

Volume of the cube = l × l × l = l³ Note:

If the length of the cube or the edge is 1 unit, then it is referred to as 1 unit cube.

●   Volume and Surface Area of Solids

Worked-out Problems on Volume of a Cuboid

7th Grade Math Problems   8th Grade Math Practice   From Volume of Cubes and Cuboids to HOME PAGE

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Volume Of Cuboid

Volume of cuboid is the total space occupied by the cuboid in a three-dimensional space. A cuboid is a three-dimensional structure having six rectangular faces. These six faces of the cuboid exist as a pair of three parallel faces. Therefore, the volume is a measure based on the dimensions of these faces, i.e. length, width and height. It is measured in cubic units.  Surface area of cuboid is the total area covered by its rectangular faces.

In this article, let us discuss what is a volume of a cuboid, its formula, along with the volume of a cuboid prism and a cube example.

What is the Volume of a Cuboid?

The volume of a Cuboid, in general, is equal to the amount of space occupied by the shape of cuboid.  It depends on the three dimensions of cuboid, i.e., length, breadth and height. The term “ Solid Rectangle ” is also known as a cuboid, because all the faces of a cuboid are rectangular.  In a rectangular cuboid, all the angles are at right angles and the opposite faces of a cuboid are equal.

Volume of Cuboid Formula

The volume of a cuboid is given by the product of its dimensions, i.e., length, width and height. The unit of volume of cuboids is cubic units or unit 3 , such as m 3 , cm 3 , in 3 , etc.

Volume of cuboid is equal to product of its base area and height. Hence, we can write;

Volume of cuboid = Base area × Height  [Cubic units]

The base of the cuboid is rectangle in shape. So, the base area of a cuboid is equal to the product of its length and breadth. Hence,

Volume of a cuboid = length × breadth × height    [cubic units]

Volume of a cuboid = l × b × h    [cubic units]

• b = breadth

Also, try:  Volume of Cuboid Calculator

How to Find Volume of Cuboid?

Volume of a cuboid is the space occupied by its dimensions, inside the cuboid. These dimensions are length, width and height. When the area of the faces of a cuboid is the same, we call this cuboid a cube. The area of all the faces of a cube is the same as they are all squares.

Think of a scenario where we need to calculate the amount of sugar that can be accommodated in a cuboidal box. In other words, we mean to calculate the volume of this box. The capacity of a cuboidal box is basically equal to the volume of the cuboid. Thus, if we know, the length, width and height of the cuboidal box, we can easily measure volume using the given formula:

Volume of cuboidal box = Length x Width x Height

Follow the below steps to find the volume of any cuboidal shape:

• Step 1: Check the dimensions of the given cuboid, i.e., length, width and height
• Step 2: Check if all the dimensions are of the same units, else we need to convert them into the same units
• Step 3: After we have made units the same for all the dimensions, multiply length, width and height together.
• Step 4: The obtained value is the volume of cuboids, written with cubic units.

Video Lesson on Volume of Cuboid

For More Information On Volumes of Cubes and Cuboid, Watch The Below Video:

Total Surface Area of Cuboid

The total surface area of a cuboid is equal to the sum of the areas of the six rectangular faces whereas the Lateral surface area of a cuboid equal to the sum of the four rectangular faces, in which two rectangular faces of the top and bottom faces are excluded. The formula for the total surface area and lateral surface area of a cuboid is given as:

Total Surface Area of a Cuboid = 2 (lb + hb + lh) square units

Lateral Surface Area of a Cuboid = 2h (l+b)

Now, let us discuss the volume of a cuboid in detail.

Volume of a Cuboid Prism

A cuboid prism or a rectangular prism is the same as a cuboid. It has 6 faces, 8 vertices, and 12 edges. When a cuboid prism or a rectangular prism has a rectangular cross-section. A prism is called a right prism when the angle between the base and the sides are at right angles. Also, the top and the bottom surfaces are in the same shape and size.  The volume of the cuboid prism is given as:

Volume of a cuboid prism or rectangular prism, V= length ×  breadth  ×  height (cubic units)

Volume of a Cube

Volume of cube : Cuboid in which length of each edge is equal is known as a cube. Thus,

Volume of a cube of side ‘a’ = a 3

Solved Examples on Volume of Cuboid

Question 1 :  Find the volume of a cuboid whose length = 5 cm, width = 2 cm and height = 3 cm.

Solution : Given,

length = 5 cm, width = 2 cm and height = 3 cm

By the formula, we know;

Volume of cuboid = length x width x height

= 5 cm x 2 cm x 3 cm

= 30 cu.cm.

Question 2 : Calculate the amount of air that can be accumulated in a room that has a length of 5 m, breadth of 6 m and a height of 10 m.

Solution : Amount of air that can be accumulated in a room = capacity of the room = volume of a cuboid

Volume of cuboid = l × b × h = 5 ×6 ×10 = 300 m 3

Thus, this room can accommodate the maximum of 300 m 3 of air.

Practice Questions

Find the volume of cuboid with following dimensions:

• Length = 15 cm, Breadth = 50 cm and Height = 22 cm
• Length = 7 m, Breadth = 3 m and Height = 5 m
• Length = 2 m, Breadth = 2.5 m and Height = 1.5 m
• Length = 80 cm, Breadth = 20 cm and Height = 44 cm
• Length = 1.7 m, Breadth = 1.5 m and Height = 1 m

To learn and practice more problems in the surface area of cuboid, you can visit BYJU’S – The Learning App

Frequently Asked Questions on Volume of Cuboid

What is the formula for volume of cube and cuboid, how do we define volume of cuboid, does the order of cuboid matters to calculate the volume, find the volume of cuboid if length = 14cm, width = 50cm and height = 10cm., if the units of dimensions of cuboid are different, then how to find the volume.

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Volume of a Cuboid

A cuboid is a 3 dimensional shape. To work out the volume we need to know 3 measurements.

The volume is found using the formula:

Volume = Length × Width ×  Height

Which is usually shortened to:

V = l × w × h

Or more simply:

In Any Order

It doesn't really matter which one is length, width or height, so long as you multiply all three together.

Example: Lengths in meters (m):

The volume is:

10 m × 4 m × 5 m = 200 m 3

It also works out the same like this:

4 m × 5 m × 10 m = 200 m 3

Try It Yourself

Volume of a Cuboid Textbook Exercise

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SOLVING WORD PROBLEMS ON CUBE AND CUBOID

Problem 1 :

Both cuboids below have the same volume. Find the height of cuboid B.

Volume of a cuboid = (length × breadth × height) cubic units

20 × 3 × 15 = 25 × 9 × h

900 = 225 × h

h = 900/225

So, the height of cuboid B is 4 cm.

Problem 2 :

The volume of the cube is twice the volume of the cuboid. Find the length of the cuboid.

Volume of a cube = (side) 3

Volume of a cuboid = y × 4 × 4

Volume of a cuboid = 16y

The volume of the cube is twice the volume of the cuboid.

Volume of a cube = 2 × Volume of a cuboid

216 = 2 × 16y

So, the length of the cuboid is 6.75 cm.

Problem 3 :

The cuboid container below is used to store boxes. Each box is a cube with side length 1m. How many boxes can be stored in the container ?

Volume of a cube = 1

Volume of a cuboid = 5 × 12 × 2

Volume of a cuboid = 120

So, 120 boxes can be stored in the container boxes.

Problem 4 :

The volume of a cuboid is 15000 cm 3 . If the length is 30 cm and the width is 25 cm, find the height of the cuboid.

volume of a cuboid = 15000 cm 3

length of cuboid = 30 cm

width of cuboid =  25 cm

height of the cuboid = ?

15000 = 30 × 25 × h

15000 = 750h

h = 15000/750

So, the height of the cuboid is 20 cm.

Problem 5 :

Shown is a net of a cuboid. Calculate the volume of the cuboid

From the given net diagram,

Length of cuboid = 24 cm

Width = 16 cm

height = 12 cm

Volume of cuboid = length x width x height

= 24 x 16 x 12

= 4608 cm 3

Problem 6 :

Find the surface area of a box with length 12 inches and width and height both 4 inches each.

Surface area = 2(l w + w h + h l)

Length = 12 inches, width = height = 4 inches

= 2 (12 x 4 + 4 x 4 + 4 x 12)

= 2 (48 + 16 + 48)

= 224 inches 2

Problem 7 :

Find the surface area of the shown below.

Area of the top = 32 cm 2

From the given figure, width = 4 cm and height = 6 cm

length x width = 32

length x 4 = 32

length = 32/4 ==> 8 cm

Surface area of rectangular prism = 2(lw + wh + hl)

= 2 (8 x 4 + 4 x 6 + 6 x 8)

= 2(32 + 24 + 48)

Problem 8 :

Find the surface area of cube.

By observing the measures, it is cube.

Side length of cube = 6 inches

Surface area of cube = 6a 2

Problem 9 :

The volume of a cuboid is 15,000 𝑐𝑚 3 . If the length is 30 cm and the width is 25 cm, find the height of the cuboid

Volume of cuboid =  15,000 𝑐𝑚 3

length = 30 cm, width = 25 cm and height = h

Length x width x height = 15,000 𝑐𝑚 3

30 x 25 x h = 15,000

h = (15000) / (30 x 25)

Problem 10 :

The ratio of the width of a cuboid to its height is 4:5. Its width is 40 cm. The ratio of the height to the length is 2:3. Find the volume of the cuboid.

Width of the cuboid = 4x, height = 5x

Width = 40 cm

Width = 4x = 4(10) ==> 40 cm

Height = 5x = 5(10) ==> 50 cm

ratio between height to the length = 2 : 3

2y = height and length = 3y

Applying the value of y, we get

Length = 3y = 3(25) ==> 75 cm

Volume of the cuboid = length x width x height

= 75 x 40 x 50

= 150000 cm 3

Problem 11 :

The two cuboids shown below have the same volume. Calculate the value of 𝑥.

x(x) (10) = 20 x 4 x 8

10x 2 = 640

So, the value of x is 8 cm.

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Worksheet on Volume of a Cube and Cuboid | Volume of Cube and Cuboid Worksheet PDF

We all are familiar with the definition of volume i.e. it is the amount of space occupied by an object. In this article, you will find various questions asked on the volume of cube and cuboid. Worksheet on Volume of Cube and Cuboid has got a plethora of questions ranging from easy to hard levels. You can use them to quickly understand the concept as well as learn how to solve word problems on the Volume of Cubes and Cuboids in no time. Practice the Problems on Cube and Cuboid Volume and attempt the exam with utmost confidence and clear with flying colors.

• Worked-out Problems on Volume of a Cuboid

Volume of a Cube and Cuboid Questions and Answers

Example 1. Find the volume of a cube whose each edge is (i) 4 cm (ii) 6 cm (iii) 8 cm

We know the volume of the cube v=s 3 where s is the edge length of the cube. (i) Given edge s=4 cm volume=4 3 =64 Hence, the volume of the cube is 64 (ii) Given,edge s=6 cm volume=6 3 =216 Hence, the volume of the cube is 216 (iii) Given, edge s=8 cm volume=8 3 =512 Hence, the volume of the cube is 512.

Example 2. Find the volume of the cuboid whose dimensions are (i) length=8 inches, breadth=5 inches, height=3 inches (ii) length=12 inches,breadth=15inches , height=13 inches (iii) length=23 inches,breadth=12 inches, height=18 inches

(i) Given, The length of the cuboid=8 inches The breadth of the cuboid=5 inches The height of the cuboid=3 inches we know that volume of the cuboid v=lbh volume v=(8.5.3)in 3 =120 in 3 Hence, the volume of the cuboid is 120 in 3 . (ii) Given, The length of the cuboid=12 inches The breadth of the cuboid=15 inches The height of the cuboid=13 inches we know that volume of the cuboid v=lbh volume v=(12.15.13)in 3 =2340 in 3 Hence, the volume of the cuboid is 2340 in 3 . (iii) Given, The length of the cuboid=23 inches The breadth of the cuboid=12 inches The height of the cuboid=18 inches we know that volume of the cuboid v=lbh volume v=(23.12.18)in 3 =4968 in 3 Hence, the volume of the cuboid is 4968 in 3 .

Given, Edge of the cube=6 cm The Length of the cuboid=5 cm The breadth of the cuboid=3 cm The height of the cuboid=7 cm We know that volume of the cube= edge 3 =6 3 =216 cm 3 As we know the volume of the cuboid=lbh =5 cm.3 cm.7 cm =105 cm 3 Hence, the volume of the cube has more volume.

Given, Length of the shelf=30 cm width of the shelf=40 cm height of the shelf=50 cm The volume of the shelf=l.b.h =30 cm.40 cm.50 cm =60,000 cm 3 Hence, the volume of Harish’s shelf is 60,000 cm 3 .

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Example 5. Water pond is 40 meters long,15 meters in width, and 10 meters in height. Find the capacity of the water pond in cubic meters?

Given, Length of the water pond=40 meters width of the water pond=15 meters height of the water pond=10 meters The capacity of the water pond=l.b.h =40.15.10 m 3 =6,000 m 3 1000 lit=1 m 3 =6,000(1000) =60,00,000 liters Hence, the capacity of the water pond is 60,00,000 liters.

Example 6. The volume of the container is 3000 m 3 . The length and breadth of the container are 20 m and 10 m respectively. Find the height of the container?

Given, The volume of the container=3000 m 3 The length of the container=20 m The breadth of the container=10 m The height of the container=? We know that volumr=lbh Let the height of the container be x. 20.10.x=3000 200x=3000 x=3000/200 =15 m Therefore, the height of the container is 15 m.

Example 7. A cubical water tank can hold 512000 liters of water. Find the length of its side in meters?

The volume of the water tank=512000 liters 1 m 3 =1000 liters The volume of the tank in m 3 = 512000/1000 a 3 =512 a=8 Hence, the length of the side of the water tank is 8 meters.

Given, metallic cube side=10 cm length of the cuboid=5 cm height of the cuboid=10 cm The volume of cube=volume of cuboid a 3 =10 3 Volume of cube=1000 length.breadth.height=1000 5 .breadth.10 =1000 breadth=1000/50=20 m Hence, breadth of the cuboid is 20 meters.

Given, side of the cube=5 cm Volume of cube = side 3 = 5 3 =125 The volume of the cube is 125 cubic cm.

Given, The length, breadth, height of the wooden box are in the ratio 6: 5 : 3 Volume=11250 cm 3 Let 6x,5x,3x be the length, breadth, and height of a wooden box. 6x.5x.3x=11250 90x 3 =11250 x 3 =11250/90=125 x=5 cm length=6x=6(5)=30 cm breadth=5x=5(5)=25 cm height=3x=3(3)=9 cm Hence, the length,breadth,and height of the wooden box are 30cm,25cm,9cm.

Given, The dimensions of the cuboid are 2cm,25 cm, and 5 cm. The volume of the cuboid=lbh =2.25.5 =250 cm 3 Also given, The volume of a cuboid is twice the volume of a cube. The volume of cube=volume of cuboid/2 =250/2=125 cm 3 The volume of the cube=a 3 a 3 =125 a=5 The total surface area of the cube=6a 2 =6(5)2 =150 cm 2 Hence, the total surface area of the cube is 150 cm 2 .

Let each edge of a cube =a cm and length bredth and heught of a cuboid are l ,b and h cm. Acordingly Volume of cube= a 3 =125 cm 3 a=5cm…………………(1) Volume of cuboid =l×b×h cm 3 l.b.h =50 cm 3 ………………(2) Area of base of cube =a^2 cm 2 Area of base of cuboid =l×b cm 2 . l.b=a 2 , put a=5 cm from eq.(1). l.b=(5 cm) 2 l.b =25 cm 2 …………………..(3) Divide eq.(2) by eq (3) l.b.h/l.b =50 cm 3 /25 cm 2 h = 2 cm Hence, the height of the cuboid is 2 cm.

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VOLUME OF CUBOID WORD PROBLEMS

If the length, width and height of a cuboid are l, w and h respectively, then

volume  of the cuboid = l x w x h cubic units

Problem 1 :

The length, breadth and depth of a pond are 20.5 m, 16 m and 8 m respectively. Find the capacity of the pond in liters.

l = 20.5 m, w = 16 m, h = 8 m

Capacity of pond :

= l x w x h

= 20.5 (16) (8)

1 m 3  = 1000 liters

= 2624(1000) liters

= 2624000 liters

Problem 2 :

The dimensions of a brick are 24 cm x 12 cm x 8 cm. How many such bricks will be required to build a wall of 20 m length, 48 cm breadth and 6 m height?

Volume of 1 brick :

= 24(12)(8)

  = 2304 cm 3

Dimensions of the wall :

l = 20 m = 2000 cm

h = 6 m = 6000 cm

Volume of wall :

= 2000(48)(600)

= 57600000 cm 3

Number of bricks required :

= 57600000/2304

Problem 3 :

The volume of a container is 1440 m 3 . The length and width of the container are 15 m and 8 m respectively. Find its height

Volume of container = 1440 m 3

length x width x height = 1440

15 x 8 x h = 1440

height = 1440/120

height = 12 m

Problem 4 :

The side of a metallic cube is 12 in. It is melted and formed into a cuboid whose length and width are 18 in and 16 cm respectively. Find the height of the cuboid.

volume of cuboid = volume of cube

l x w x h = a 3

18 x 16 x h = 12 3

288h = 1728

Divide both sides by 288.

Problem 5 :

The length, width and height of a cuboid are in the ratio 7 : 5 : 2. Its volume is 35840 cm3. Find its dimensions.

From the ratio 7 : 5 : 2, the dimensions of the cuboid are

length = 7x, width = 5x, height = 2x

Volume = 35840 cm 3

(7x)(5x)(2x) = 35840

70x 3  = 35840

Divide each side by 70.

x 3  = 512

x 3  = 8 3

Length = 7(8) = 56 cm

Width = 5(8) = 40 cm

Height = 2(8) = 16 cm

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Volume of Cube

The volume of a cube is defined as the total number of cubic units occupied by the cube completely. To calculate the volume of a cube, we need to know the length of any one side of the cube. The formula for finding the volume of cube is: Volume = side x side x side.

A cube is a three-dimensional solid figure, having 6 square faces. Volume is nothing but the total space occupied by an object. An object with a larger volume would occupy more space. Let us understand the volume of a cube in detail along with the formula and solved examples in the following sections.

What is Volume of Cube?

The volume of a cube is the total three-dimensional space occupied by a cube. A cube is a 3-D solid object with six square faces, having all the sides of the same length. The cube is also known as a regular hexahedron and is one of the five platonic solid shapes .

The unit of volume of the cube is given as the (unit) 3 or cubic units. The SI unit of volume is the cubic meter (m 3 ), which is the volume occupied by a cube with each side measuring 1m. The USCS units for volume are inches 3 , yards 3 , etc.

Volume of Cube Formula

We can calculate the volume of any cube using different formulas based on the given parameters. It can be calculated using the side length or the measure of the cube's diagonal.

• Volume of a cube (V) whose side is s is, V = s 3 .
• Volume of the cube (V) whose diagonal is d is, V = (√3×d 3 )/9

Volume of Cube Using Side Formula

The volume of a cube can be found by multiplying the edge length three times by itself. i.e., the volume of a "cube" is obtained by " cubing " the side length. For instance, if the length of an edge of a cube is 4, the volume will be 4 3 . The formula to calculate the volume of a cube is given as,

Volume of a cube = s 3 , where 's' is the length of the side of the cube.

The concept to obtain the volume of a cube formula can be understood using the following steps,

• Consider any square-shaped sheet of paper.
• Now, the area covered by this square sheet will be its surface area i.e. its length multiplied by its breadth. As for a square, since the length and breadth are equal, the surface area will be “s 2 “.
• A cube is made by stacking multiple square sheets on top of each other so that the height becomes equal to the length and breadth, i.e., “s” units.
• This gives us the height or thickness of the cube as “s”.
• It can thus be concluded that the overall space covered by the cube, which is the volume, will be the area of the base multiplied by the height. i.e., s 2 × s = s 3 .

Volume of Cube Using Diagonal Formula

The volume of the cube can also be found out directly by another formula if the diagonal is known.

The diagonal (d) of a cube is given as, d = 3s, where, 's' is the side length of the cube. From this formula, we can write 's' as, s = d/√3.

By substituting this in the above formula (V = s 3 ), we get:

V = (d/√3) 3 = d 3 /(3√3)

By rationalizing the denominator :

V = (√3d 3 )/9

Thus, the volume of a cube equation using diagonal can finally be given as:

Volume of the cube = (√3×d 3 )/9, where d is the length of the diagonal of the cube.

Note: A common mistake is to be avoided by not confusing the diagonal of a cube with the diagonal of its face. The diagonal of a cube cut through its center, as shown in the figure above. While the face diagonal is the diagonal on each face of the cube.

How To Find the Volume of a Cube?

The volume of a cube can be easily found out by just knowing the length of its edge or the measure of its diagonal. Different steps to be followed to calculate the area of the cube depending upon the given parameters will be covered in this section.

Finding Volume of Cube Given Edge Length

The measure of all the sides of a cube is the same thus, we only need to know one side in order to calculate the volume of the cube. The steps to calculate the volume of a cube using the side length are,

• Step 1: Note the measurement of the side length of the cube.
• Step 2: Apply the formula to calculate the volume using the side length: Volume of cube = (side) 3 .
• Step 3: Express the final answer along with the unit (cubic units) to represent the obtained volume.

Example: Calculate the volume of a cube with a side length of 2 inches.

Solution: The volume of a cube with a side length of 2 inches would have a volume of (2 × 2 × 2) = 8 cubic inches.

Thus, it can hold a total of 8 cubes of 1 inch each. The same can be understood with the help of the given diagram.

Calculating Volume of Cube Given Diagonal

Given the diagonal, we can follow the steps given below in order to find the volume of a given cube.

• Step 1: Note the measurement of the diagonal of the given cube.
• Step 2: Apply the formula to find the volume using diagonal: [√3×(diagonal) 3 ]/9
• Step 3: Express the obtained result in cubic units.

Example: Find the volume of a cube with the diagonal measuring 3 in.

Given: Diagonal = 9 in

We know, volume of cube = [√3×(diagonal) 3 ]/9 ⇒ Volume = [√3×(3) 3 ]/9 = 3 × √3 = 3 × 1.732 = 5.196 in 3 .

Important Notes on Volume of Cube:

The formulas to find the volume of a cube are:

• V = s 3 , where s is the edge length of the cube.
• V = √3×d 3 /9, where d is the diagonal length of the cube.

Challenging Questions:

• If the side dimensions of the two cubes are 8 inches and 12 inches, how many small cubes can fit in the larger one?
• Why will the ratio of the volume of two cubes with side lengths in the ratio of 1:2 be 1:8?

☛ Related Topics:

• Volume of Cubical Box
• Surface Area of Cube

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Volume of Cube Examples

Example 1: Using the volume of a cube formula, calculate the side length of a Rubik's cube whose volume is 64 in 3 .

Solution: To find: Length of the cube (s) = 4 inches

Given: Volume of Rubik's cube = 64 in 3

Using the volume of cube formula,

Volume of the cube = s 3 , where 's' is the side length.

Putting the values, we get,

⇒ 64 = (s 3 )

⇒ s = (64) 1/3 = 4 in

Answer: Side length of the Rubik's cube = 4 in.

Example 2: Find the volume of a cube if the length of its diagonal is 12 inches?

Solution: To find: Volume of cube

Given: Diagonal of the cube = 12 in.

The volume of the cube equation when diagonal is given is:

Volume of cube = (√3×d 3 )/9

⇒ Volume of given cube = (√3×12 3 )/9 = 332.553 in 3

Answer: Volume of the cube = 332.553 in 3

Example 3: A cube with an edge length of 6 cm and a cuboid with dimensions 6 cm × 5 cm × 8 cm are on a table. Which shape has more volume?

The volume of the cube = 6 3 = 216 cm 3 .

The volume of the cuboid = 6 cm × 5 cm × 8 cm = 240 cm 3

Answer: Cuboid has more volume.

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Practice Questions on Volume of Cube

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FAQs on Volume of Cube

What is the definition of volume of cube.

The volume of a cube is defined as the total space enclosed by the cube in a three-dimensional space. It represents the total number of cubic units completely occupied by the cube. The volume of a cube helps in determining the capacity of a cubical-shaped object.

What Is the Formula for the Volume of a Cube?

The volume of a cube is obtained by multiplying its side three times. The formula of volume of the cube can thus be given as, Volume of cube = s 3 , where s is the side length of the cube.

How To Calculate the Volume of the Cube?

To calculate the volume of a cube, we either need the measurement of its side length or the length of its diagonal.

• To find the volume using the side length of a cube, we multiply the side three times.
• In order to calculate a cube's volume using diagonal, we can apply the formula: (√3×d 3 )/9, where d is the length of the cube's body diagonal.

What Is the Unit of Volume of the Cube?

The unit of volume of a cube is given as the cubic units or (unit) 3 . Also, the SI unit of volume is the cubic meter (m 3 ), which is the volume occupied by a cube with each side measuring 1m. Some other important units are cubic feet(ft 3 ), cubic centimeters(cm 3 ), cubic millimeters(mm 3 ), cubic inches(in 3 ), cubic yards(yd 3 ), etc.

What are the Formulas for Finding the Volume of Cube Cuboid and Cylinder ?

• The volume of cube = (side) 3
• The volume of cuboid = length × width × height
• The volume of cylinder = π(radius) 2 (height)

How To Find the Side of a Cube When Given the Volume?

The volume of a cube equation using its side is as V = side × side × side or (side) 3 . This formula can be rearranged to calculate the side length as side = ∛V. Here, the symbol ∛ stands for cube root .

What Is the Volume of Cube With Side 1 Meter?

To find the volume of a cube, we find the cube of its side length. The volume of a cube with the side measuring 1 meter = (1) 3 m 3 = 1 m 3 . This value represents the total space enclosed by the given cube.

How to Find Volume of Cube Using Calculator?

Volume of a cube can be easily and quickly determine with the help of the volume of a cube calculator. It is an online tool that helps children to do the calculation with accuracy and get the answers within seconds. To find the volume of a cube using a calculator we required sufficient data or the value of certain parameters such as measurement of the edge of a cube. Try Cuemath's volume of a cube calculator and get your answers just by a click.

☛Check volume of cubes worksheets now for more practice.

How To Find Volume of Cube If Diagonal is Given?

To find the volume of a cube, given the diagonal, we can apply the formula: (√3×d 3 )/9, where d is the length of the cube's body diagonal. Remember that this formula is applicable when the length of the body diagonal is given and not the face diagonal.

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Volume of Cube & Cuboid (Problem Solving)

Subject: Maths for early years

Age range: 3-5

Resource type: Worksheet/Activity

Last updated

20 January 2015

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Very attractive slides. Thank you

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