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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:

Substitute the values into the formula.

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

Work out the calculation.

Write the answer, include the units.

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.

Substitute the values that you do know into the formula.

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}

Solve the equation.

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:

• Know and apply the formula to calculate the volume of cuboids
• Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3-D

The next lessons are

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

Still stuck?

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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.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

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Volume of Cuboid

The volume of cuboid is the quantity that is used to measure the space in a cuboid. A cuboid is a three-dimensional shape that can be seen around us very often. The term volume is used in measuring the capacity of any shape based on its dimensions such as length, breadth, and, height. To calculate the volume of a cuboid, a formula specific to the shape of a cuboid will be used. In this section, we will be learning the formula to calculate the volume of cuboid and solve a few examples to understand the concept better.

What is the Volume of Cuboid?

The volume of the cuboid is the measure of the space occupied within a cuboid. The cuboid is a three-dimensional shape that has length, breadth, and height. If we have a rectangular sheet and we go on stacking such sheets, we will end up getting a shape that has some length, breadth, and height. This stack of sheets looks like a shape that has 6 faces, 12 edges, and 8 vertices hence giving us the shape of a cuboid. The unit of volume of cuboid is given as the (unit) 3 . The metric units of volume are cubic meters or cubic centimeters while the United States Customary System (USCS) units of volume are, cubic inches or cubic feet. The volume of the cuboid depends on the length, breadth, and height of the cuboid, hence changing any one of those quantities changes the volume of the shape.

Volume of Cuboid Formula

The formula for the volume of the cuboid can be derived from the concept explained on rectangular sheets. Let the area of a rectangular sheet of paper be 'A', the height up to which they are stacked be 'h' and the volume of the cuboid be 'V'. Then, the volume of the cuboid is given by multiplying the base area and height. The volume of cuboid = Base area × Height The base area for cuboid = l × b Hence, the volume of a cuboid, V = l × b × h = lbh

How to Calculate Volume of Cuboid?

The volume of a cuboid is the space occupied inside a cuboid. If all the three dimensions of a cuboid get equal, it becomes a cube . The volume of the cuboid can be calculated using the formula of the volume of the cuboid. The steps to calculate the volume of a cuboid are:

• Step 1: Check if the given dimensions of cuboids are in the same units or not. If not, convert the dimensions into the same units.
• Step 2: Once the dimensions are in the same units, multiply the length, breadth, and height of the cuboid.
• Step 3: Write the unit in the end, once the value is obtained.

Let us take an example to learn how to calculate the volume of a cuboid using its formula.

Example: Find the volume of the cuboid having a length of 7 inches, breadth of 5 inches, and height of 2 inches. Solution: As we know, the volume of a cuboid, V = lbh Here, length l = 7 inches, breadth b = 5 inches and height h = 2 inches Thus, volume of cuboid, V = lbh = (7 × 5 × 2) in 3 ⇒ V = 70 in 3 \(\therefore\) The volume of cuboid is 70 in 3 .

Volume of Cuboid Examples

Example 1: If the dimensions of a cuboidal fish aquarium are, 30 inches, 20 inches, and 15 inches. Can you determine the volume of the fish aquarium? Solution: As we know, the fish aquarium is of cuboidal shape. Hence, the dimensions of the fish aquarium are: Length of the aquarium = 30 in Width of the aquarium = 20 in Height of the aquarium = 15 in

The volume of the aquarium is given as: Volume = Length × Width × Height ⇒ Volume = 30 × 20 × 15 in 3 = 9000 in 3 ∴ The volume of the cuboidal fish aquarium is 9000 cubic inches.

Example 2: What will be the length of the cuboid if its volume is 3000 in 3 , breadth is 15 inches and height is 10 inches? Solution: As we know, volume of a cuboid is given as Volume = Length × Breadth × Height. The given dimensions for cuboid are: Volume = 3000 in 3 Breadth = 15 in Height = 10 in Let the length of cuboid is x inches.

Hence, the volume of the cuboid will be: Volume = Length × Breadth × Height ⇒ Volume = x × 15 × 10 = 3000 in 3 ⇒ x = (3000/(15 × 10)) = 20 in ∴ The length of cuboid is 20 inches.

Example 3: Find the volume of cuboid if the length is 10 in, breadth is 20 in, and height is 30 in.

Solution: volume of a cuboid is given as Volume = Length × Breadth × Height. The given dimensions for cuboid are: Length = 10 in Breadth = 20 in Height = 30 in ⇒ Volume = 10 × 20 × 30 in 3 = 6000 in 3 ∴ The volume of the cuboid is 6000 cubic inches.

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Practice Questions on Volume of Cuboid

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FAQs on Volume of Cuboid

What do you mean by volume of cuboid.

The volume of a cuboid is the space that is enclosed within a cuboid. For example, in order to fill water in an aquarium, we must know its volume.

How to Find the Volume of Cuboid?

The volume of a cuboid is calculated by multiplying its length, width, and height. For example, the volume of a cuboid of length, width, and height 2 inches, 3 inches, and 4 inches is given as Volume = length × width × height = 2 × 3 × 4 = 24 inch 3

What Is the Formula for the Volume of Cuboid?

The formula of volume of a cuboid is = Length × Width × Height. The formula for the volume of cuboid is deduced by stacking rectangular sheets one over another thereby giving us three parameters in the formula length, width, and height.

If the Units of Dimensions of a Cuboid Are Different, Then How to Find the Volume of Cuboid?

If the units of the given dimensions of a cuboid are different, then first we would need to change the units of dimensions of any two dimensions in the unit of the third dimension. After that, multiply all three dimensions known to us, to calculate the volume of the cuboid.

Does the Order of Height, Width, and Length Matter While Calculating the Volume of Cuboid?

No, the order of height, width, and length does not matter while finding a cuboid's volume because we need to multiply all three quantities for determining it. As multiplication is associative, hence, no matter in whichever order dimensions are multiplied the volume of the cuboid remains the same.

How Does the Volume of Cuboid Change When the Length of Side Is Halved?

The volume of the cuboid gets half when the length of the side is halved as l = l/2. As, volume of cuboid = length × width × height = (l/2)× b × h = (lbh)/2 = volume/2. Thus, the volume of the cuboid gets halved as soon as its length gets halved.

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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 \(c = 3\).

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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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Volume of Cuboids Differentiated Worksheet with Solutions

Subject: Mathematics

Age range: 11-14

Resource type: Worksheet/Activity

Last updated

1 December 2018

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Volume of cubes and cuboids: Differentiated problem solving questions (Mild to mega spicy)

There are 23 questions with solutions attached

Mild: 4 easy worded questions Hot: 5 questions that involve converting units Very Spicy: 4 questions for finding missing sides Super Spicy: 6 problem solving worded questions including using volume to find missing side then find SA. Mega spicy: More challenging questions that require good literacy and numeracy skills Cross Curricular : A worded science problem involving converting units of volume.

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Volume of Cuboid

Here we will learn how to solve the application problems on Volume of cuboid using the formula.

Formula for finding the volume of a cuboid

Volume of a Cuboid (V) = l × b × h;

Where l = Length, b = breadth and h = height.

1. A field is 15 m long and 12 m broad. At one corner of this field a rectangular well of dimensions 8 m × 2.5 m × 2 m is dug, and the dug-out soil is spread evenly over the rest of the field. Find the rise in the level of the rest of the field.

The volume of soil removed = The Volume of the Well

                                         = 8 m × 2.5 m × 2 m

                                         = 8 × 2.5 × 2 m 3

                                         = 40 m 3

Let the level of the rest of the field be raised by h.

The volume of the soil spread evenly on the field

                            = Volume of the cuboid of dimensions + Volume of the cuboid of dimensions

                            = 2.5 m × 4 m × h + 12.5 m × 12 m × h

                            = (2.5 m × 4 m × h + 12.5 m × 12 m × h)

                            = (10h + 150h) m\(^{2}\)

                            = 160h m\(^{2}\)

Therefore, 160h m\(^{2}\) = 40 m 3

⟹ h = \(\frac{40}{160}\) m

⟹ h = \(\frac{1}{4}\) m

Therefore, the rise in the level = \(\frac{1}{4}\) m

                                            = 25 cm

2. Squares each side 8 cm are cut off from the four corners of a sheet of tin measuring 48 cm by 36 cm. The remaining portion of the sheet is folded to form a tank open at the top. What will be the capacity of the tank?

To make the tank, NGHP has to folded up along NP, LMQK along MQ, EFNM along MN and IJQP.

Now, MN = QP = (48 - 2 × 8) cm = 32 cm, and

NP = MQ = (36 - 2 × 8) cm = 20 cm.

EM = KQ = IP = GN = 8 cm.

Therefore, the capacity of the tank = 32 × 20 × 8 cm 3

                                                   = 5120 cm 3

                                                   = 5.12 litres [Since, 1 litre = 1000 cm 3 ]

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Worked-out Problems on Volume of a Cuboid | How to Find Cuboid Volume?

Students who want to learn the volume of cuboids and cubes can use the Worked-out Problems on Volume of a Cuboid here. Get to see various examples on the cuboid volume in the coming sections. Try to solve the questions and improve your preparation standards. Check out the Cube and Cuboid Word Problems with solutions in the below sections.

Question 1.

Find the volume of a cuboid of length 18 cm, breadth 25 cm, and height 5 cm?

Given that,

Length of cuboid = 18 cm

Breadth of cuboid = 25 cm

Height of cuboid = 5 cm

Cuboid volume = length x breadth x height

Volume = 18 x 25 x 5

Therefore, the volume of a cuboid is 2250 cm³.

Question 2.

If the area of the base and height of the cuboid is 212 cm², 8 cm, calculate cuboid volume?

The base of the base = 212 cm²

Height of a cuboid = 8 cm

Cuboid volume = (Area of the base) x height

Volume = 212 x 8 = 1696 cm³.

Question 3.

Find the volume of the cube whose each side is 16 cm?

Side length of cube a = 16 cm

The volume of the cube V = a³

V = 16³ = 16 x 16 x 16

V = 4096 cm³

Therefore, the cube volume is 4096 cm³.

Question 4.

If the cuboid volume is 512 cm³, its length and height is 8 cm, 7 cm. Find the cuboid breadth?

Cuboid Volume = 512 cm³

Cuboid length = 8 cm

Cuboid height = 7 cm

Cuboid breadth = Volume / (length) x (height)

= 512 / (8 x 7)

= 512 / 56 = 9.142 cm

Therefore, the breadth of cuboid is 9.142 cm.

Question 5.

The length, breadth, and depth of a lake are 15 m, 20 m, 9 m respectively. Find the capacity of the lake in liters?

Length of lake = 15 m

Breadth of lake = 20 m

Depth of lake = 9 m

Capacity of lake = (length) x (breadth) x (depth)

= 15 x 20 x 9 = 2700 m³

1000 liter = 1 m³

Capacity of lake in Litres = 2700 x 1000

= 2700000 litres

Therefore, the capacity of lake is 2700000 litres.

Question 6.

The dimensions of the brick are 25 cm x 8 cm x 10 cm. How many such bricks are required to build a wall of 16 m in length, 20 cm breadth, and 8 m in height?

Length of brick = 25 cm

Breadth of brick = 8 cm

Height of brick = 10 cm

Length of wall = 16 m

Breadth of wall = 20 m

Height of wall = 8 m

Volume of 1 brick = length x breadth x height

= 25 x 8 x 10 = 2000 cm³

Volume of wall = length x breadth x height

= 16 x 20 x 8

= 2560 = 2560 x 100²

Number of bricks required = (2560 x 100²) / 2000

= 1280 x 10 = 12800

So, the required number of bricks are 12800.

Question 7.

External dimensions of a wooden cuboid are 20 cm × 15 cm × 12 cm. If the thickness of the wood is 2 cm all around, find the volume of the wood contained in the cuboid formed.

External length of cuboid = 20 cm

External breadth of cuboid = 15 cm

External height of the cuboid = 12 cm

External volume of the cuboid = (length x breadth x height)

= (20 x 15 x 12) = 3600 cm³

Internal length of cuboid = 20 – 4 = 16 cm

Internal breadth of cuboid = 15 – 4 = 11 cm

Internal height of the cuboid = 12 – 4 = 8 cm

Internal volume of a cuboid = (length x breadth x height)

= (16 x 11 x 8) = 1408 cm³

Therefore, volume of wood = External volume of the cuboid – Internal volume of a cuboid

= 3600 – 1408 = 2192 cm³

∴ Volume of the wood contained in the cuboid is 2192 cm³.

Question 8.

The volume of a container is 1440 m³. The length and breadth of the container are 15 m and 8 m respectively. Find its height?

Length of the container = 15 m

The breadth of the container = 8 m

The volume of the container = 1440 m³

(length x breadth x height) = 1440

15 x 8 x height = 1440

120 x height = 1440

height = 1440/120

height = 12

∴ The height of the container is 12 m.

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Volume of cubes and cuboids

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Volume of a Cube Worksheets

Walk through this unit of printable volume of a cube worksheets hand-picked for 5th grade, 6th grade, and 7th grade students, comprising pdfs with problems presented as solid shapes and as word problems. A cube is a unique 3-dimensional shape that has squares for all six of its sides. Know the formula for the volume of the cube and use it to solve mathematical problems. Also, learn to find the side length of the cubes. Begin your learning with our free worksheets.

Volume of Cubes | Integers - Easy

Multiply the length of the given side thrice to calculate the volume. Gain a conceptual understanding of volume and solve problems presented as 3D shapes and in word problems with dimensions involving integers ≤ 20.

• Download the set

Volume of Cubes | Integers - Moderate

Add-on to your practice and level up with this batch of printable volume of a cube worksheets for grade 5. Find the volume of each cube whose side lengths are presented as 2-digit integers.

Volume of Cubes | Decimals

Plug in the measure of the side length (a) in the volume of a cube formula V = a 3 to determine the volume of the cube. The side length is expressed as decimals. Compute and round off the answer to two decimal places.

Volume of Cubes | Fractions

Convert mixed fractions to improper fractions if required and then multiply side length thrice presented as fractions to figure out the volume enclosed by each cube in these pdf worksheets for grade 6 and grade 7.

Finding the Side length of the Cube

Solve for the side length or the edge of the cube by rearranging the volume of the cube formula. Determine the cube root of the given volume to find the length of the side. Round off to the nearest tenth.

Related Worksheets

» Volume by Counting Cubes

» Volume of Rectangular Prisms

» Volume of Triangular Prisms

» Volume of Prisms

» Volume of Mixed Shapes

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Practice problems of the cuboid

Number of problems found: 724.

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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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SOLVING WORD PROBLEMS INVOLVING VOLUME OF CUBES

Formula : 

Volume of a cube = a 3 cubic units

Problem 1 :

Find the volume of a cube each of whose side is (i) 5 cm (ii) 3.5 m (iii) 21 cm

Volume of cube = a 3

(ii) 3.5 m :

= 42.875 cm 3

(iii) 21 cm :

= 9261 cm 3

Problem 2 :

A cubical milk tank can hold 125000 liters of milk. Find the length of its side in meters.

Volume of cubical milk tank = 125000 liters

1000 liters = 1 m 3 ,

Volume of tank = 125000/1000

a 3  = 125

Problem 3 :

A metallic cube with side 15 cm is melted and formed into a cuboid. If the length and height of the cuboid is 25 cm and 9 cm respectively then find the with of the cuboid.

volume of cuboid = volume of cube

l x w x h = a 3

25 x w x 9 = 15 3

225w = 3375

Divide each side by 225.

Problem 4 :

The sides of two cubes A and B are in the ratio 3 : 5. If the volume of cube A is 729 cm 3 , find the volume of cube B. 

From the ratio 3 : 5, the sides of cubes A and B are 

Volume of cube A = 243 cm 3

(3x) 3 = 729

27x 3  = 729

Divide each side by 27.

x 3  = 27

x 3  = 3 3

Side of cube A = 3(3) = 9 cm

Side of cube B = 5(3) = 15 cm

Volume of cube B : 

= 3375 cm 3

Problem 5 :

If the sides of two cubes are in the ratio 4 : 7, find the ratio of their volumes. 

From the ratio 4 : 7, the sides of two cubes are 

Ratio of their volumes :

= (4x) 3 : (7x) 3

= 64x 3  : 343x 3

Divide both the terms of the ratio by x 3 .

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