Volume Formula

Here we will learn about volume formulas, including the formula for calculating the volume of 3D shapes and how to use the volume formula to solve problems.

There are also 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 a volume formula?

A volume formula is a formula which is used to calculate the volume of a 3D shape. Volume is the amount of space there is inside a shape.

To calculate the volume of an object in three dimensions, we can use the various volume formulas.

Cube
Volume Formula image 1
 Volume =a3 \text { Volume }=a^3
Cuboid
(or rectangular prism)

Volume Formula image 2
Volume=l×w×h \text{Volume}=l\times w\times h
Prism
Volume Formula image 3
Volume=Area of cross section×Length \text{Volume}=\text{Area of cross section} \times \text{Length}
Cylinder
Volume Formula image 4
Volume=πr2h \text{Volume}=\pi r^2h
Pyramid
Volume Formula image 5
Volume=13×Area of base×Height \text{Volume}=\frac{1}{3}\times \text{Area of base} \times \text{Height}
Cone
Volume Formula image 6
Volume=13πr2h \text{Volume}=\frac{1}{3}\pi r^2h

(Volume=13×Area of base×Height) (\text{Volume}=\frac{1}{3}\times \text{Area of base} \times \text{Height})
Sphere
Volume Formula image 7
Volume=43πr3 \text{Volume}=\frac{4}{3}\pi r^3

The formulae for the volume of a cone and the volume of a sphere are given to you for GCSE maths. You need to learn the other formulae.

Volume is measured in cubic units.

  • Metric units examples

cm3 cm^3 (cubic centimetres),

m3 m^3 (cubic metres),

mm3 mm^3 (cubic millimetres).

  • Imperial units examples

cubic feet,

cubic inches.

Volume can also be described using units of capacity such as millilitres, litres, pints or gallons.

What is a volume formula?

What is a volume formula?

Download our free volume formula poster to focus your revision!

Volume formula poster

How to use volume formula

In order to use volume formula:

  1. Write down the formula.
  2. Substitute the values into the formula.
  3. Complete the calculation.
  4. Write the answer, including the units.

Explain how to use volume formula

Explain how to use volume formula

Volume worksheet (includes volume formula)

Volume worksheet (includes volume formula)

Volume worksheet (includes volume formula)

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

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Volume worksheet (includes volume formula)

Volume worksheet (includes volume formula)

Volume worksheet (includes volume formula)

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

DOWNLOAD FREE

Related lessons on how to calculate volume

Volume formula is part of our series of lessons to support revision on how to calculate volume and 3D shapes. You may find it helpful to start with the main 3D shapes 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:

Volume formula examples

Example 1: volume of a cylinder

Calculate the volume of the cylinder below.

Volume formula example 1 image

  1. Write down the formula.

The 3D shape is a cylinder. The formula you need to use is

Volume=πr2h. \text{Volume}=\pi r^2h.

2Substitute the values into the formula.

We need to calculate the area of a circle (the circular base) and then multiply it by the height of the shape.

The values you need to substitute into the formula are the radius of the base (r=5) (r=5) and the height of the shape (h=13). (h=13).

Volume=πr2h=π×52×13 \begin{aligned} \text{Volume}&=\pi r^2h \\\\ &=\pi \times 5^2 \times 13 \end{aligned}

3Complete the calculation.

V=325π=1021.107 V=325\pi=1021.107…

4Write the answer, including the units.

The dimensions of the cylinder were given in centimetres, so the volume (the amount of three-dimensional space inside the shape) will be in cubic centimetres (cm3). (cm^3).

V=1021 cm3 V=1021 \ cm^3 (to the nearest integer).

Example 2: volume of a sphere

Calculate the volume of a sphere with diameter 9 mm. 9 \ mm.

Write down the formula.

Show step

Substitute the values into the formula.

Show step

Complete the calculation.

Show step

Write the answer, including the units.

Show step

Example 3: missing dimension

The volume of the cuboid is 2016 cm3. 2016 \ cm^{3}.

Calculate the value of x. x.

Volume formula example 3 image

Write down the formula.

Show step

Substitute the values into the formula.

Show step

Complete the calculation.

Show step

Write the answer, including the units.

Show step

Example 4: missing dimension

The volume of the square pyramid is cuboid is 66 000 cm3. 66 \ 000 \ cm^{3}.

Calculate the height of the pyramid.

Volume formula example 4 image

Write down the formula.

Show step

Substitute the values into the formula.

Show step

Complete the calculation.

Show step

Write the answer, including the units.

Show step

Example 5: volume of a compound shape

This shape is made from a cuboid and a square based pyramid. Calculate the volume of the 3D shape.

Volume formula example 5 image

Write down the formula.

Show step

Substitute the values into the formula.

Show step

Complete the calculation.

Show step

Write the answer, including the units.

Show step

Example 6: volume of a compound shape

This shape is made from a cylinder and a cone. Calculate the volume of the 3D shape.

Volume formula example 6 image

Write down the formula.

Show step

Substitute the values into the formula.

Show step

Complete the calculation.

Show step

Write the answer, including the units.

Show step

Common misconceptions

  • Missing/incorrect units

You should always include units in your answer.

Volume is measured in cubic units (for example, mm3,cm3,m3 mm^3, cm^3, m^3 ).

  • Calculating with different units

You need to make sure all measurements are in the same units before calculating the volume. (For example, you can’t have some measurements in centimetres and some in metres).

  • Make sure you get the correct formula

Be careful to make sure you use the correct formula for the volume of the three-dimensional shape. Volume formulas can be easily confused with each other and with formulas for calculating the surface area of a three-dimensional object.

  • Don’t round off too early

It is important to not round decimals until the end of the calculation. Rounding too early can result in an inaccurate answer.

  • Leave your answer in the correct form

It is important to leave your answer in terms of π \pi if asked to.

Practice volume formula questions

1. Calculate the volume.

 

Practice volume formula question 1

471 cm3 471 \ cm^3
GCSE Quiz True

157 cm3 157 \ cm^3
GCSE Quiz False

565 cm3 565 \ cm^3
GCSE Quiz False

188 cm3 188 \ cm^3
GCSE Quiz False

The volume formula to calculate the volume of the cylinder is

 

Volume=πr2h. \text{Volume} =\pi r^2h.

 

You need to substitute in the values r=5 r=5 and h=6. h=6.

 

Volume=πr2h=π×52×6=150π=471.23 \begin{aligned} \text{Volume} &=\pi r^2h \\\\ &= \pi \times 5^2 \times 6 \\\\ &=150\pi \\\\ &=471.23… \end{aligned}

 

The volume of the cylinder is 471 cm3. 471 \ cm^3.

2. Calculate the volume of this pyramid.

 

Practice volume formula question 2

240 cm3 240 \ cm^3
GCSE Quiz False

200 cm3 200 \ cm^3
GCSE Quiz False

96 cm3 96 \ cm^3
GCSE Quiz False

80 cm3 80 \ cm^3
GCSE Quiz True

The volume formula to calculate the volume of the pyramid is

 

Volume=13×area of base×height. \text{Volume} =\frac{1}{3}\times \text{area of base}\times \text{height}.

 

You need to find the area of the base of the pyramid.

 

A=5×6=30 A=5\times 6=30

 

Volume=13×area of base×height=13×30×8=80 \begin{aligned} \text{Volume} &=\frac{1}{3}\times \text{area of base}\times \text{height} \\\\ &= \frac{1}{3}\times 30\times 8 \\\\ &=80 \end{aligned}

 

The volume of the pyramid is 80 cm3. 80 \ cm^3.

3. The volume of this 720 mm3. 720 \ mm^3. Calculate the length of the missing side.

 

Practice volume formula question 3

16 mm 16 \ mm
GCSE Quiz True

16 cm 16 \ cm
GCSE Quiz False

15 mm 15 \ mm
GCSE Quiz False

15 cm 15 \ cm
GCSE Quiz False

The volume formula to calculate the volume of the cuboid is

 

Volume=l×w×h. \text{Volume} =l\times w\times h.

 

You need to substitute in the values given, using x x for the unknown length into the volume formula.

 

Volume=l×w×h720=x×5×9720=x×45 \begin{aligned} \text{Volume} &=l\times w\times h\\\\ 720&=x\times 5\times 9\\\\ 720&=x\times 45 \end{aligned}

 

The missing length is

 

x=720÷45=16. x=720\div 45=16.

 

The missing length of the cuboid is 16 mm. 16 \ mm.

4. The volume of a cylinder is 160π cm3. 160\pi \ cm^3. The height of the cylinder is 6.4 cm. 6.4 \ cm. Find the radius of the cylinder.

6 cm 6 \ cm
GCSE Quiz False

5 cm 5 \ cm
GCSE Quiz True

7 cm 7 \ cm
GCSE Quiz False

4 cm 4 \ cm
GCSE Quiz False

The volume formula to calculate the volume of the cylinder is

 

Volume=πr2h. \text{Volume} =\pi r^2h.

 

You need to substitute in the values given into the formula

 

Volume=πr2h160π=π×r2×6.4 \begin{aligned} \text{Volume} &=\pi r^2h\\\\ 160\pi&=\pi \times r^2\times 6.4 \end{aligned}

 

The workings for the missing length are

 

r2=160ππ×6.4r2=25r=25r=5 \begin{aligned} r^2&=\frac{160\pi}{\pi \times 6.4}\\\\ r^2&=25\\\\ r&=\sqrt{25}\\\\ r&=5 \end{aligned}

 

The radius of the cylinder is 5 cm. 5 \ cm.

5. Calculate the volume of the shape below.

 

Practice volume formula question 5

1150 cm3 1150 \ cm^3
GCSE Quiz False

1400 cm3 1400 \ cm^3
GCSE Quiz False

1200 cm3 1200 \ cm^3
GCSE Quiz True

2600 cm3 2600 \ cm^3
GCSE Quiz False

The volume of the cuboid is

 

Volume=l×w×h=10×10×10=1000 \begin{aligned} \text{Volume}&=l\times w\times h\\\\ &=10\times 10\times 10\\\\ &=1000 \end{aligned}

 

Or you can take one side of the cube and cube it.

 

103=1000 10^3=1000

 

The volume of the pyramid is

 

Volume=13×area of base×height=13×10×10×6=200 \begin{aligned} \text{Volume}&=\frac{1}{3}\times \text{area of base}\times \text{height}\\\\ &=\frac{1}{3}\times 10\times 10\times 6\\\\ &=200 \end{aligned}

 

The total volume can be found by adding the two volumes together.

 

 

Total Volume=1000+200=1200 cm3 \text{Total Volume}=1000+200=1200 \ cm^3

6. Calculate the volume of the shape below. Give your answer to 3 3 significant figures.

 

Practice volume formula question 6

516π cm3 516\pi \ cm^3
GCSE Quiz True

660π cm3 660\pi \ cm^3
GCSE Quiz False

684π cm3 684\pi \ cm^3
GCSE Quiz False

1116π cm3 1116\pi \ cm^3
GCSE Quiz False

The volume of the cylinder is

 

Volume=πr2h=π×62×12=432π \begin{aligned} \text{Volume}&=\pi r^2h\\\\ &=\pi \times 6^2\times 12\\\\ &=432\pi \end{aligned}

 

The volume of the cone is

 

Volume=13πr2h=13×π×62×7=84π \begin{aligned} \text{Volume}&=\frac{1}{3}\pi r^2h\\\\ &=\frac{1}{3}\times \pi \times 6^2\times 7\\\\ &=84\pi \end{aligned}

 

The total volume can be found by adding the two volumes together.
Total Volume=432π+84π=516π cm3 \text{Total Volume}=432\pi+84\pi=516\pi \ cm^3

Volume formula GCSE questions

1. Shown below is a prism.

The cross-sectional area is 16 cm2. 16 \ cm^2.

The prism has length 11 cm. 11 \ cm.

 

Find the volume of the prism.

Include the units.

 

Volume formula GCSE question 1

 

(3 marks)

Show answer
16×11 16 \times 11

(1)

176 176

(1)

cm3 cm^3

(1)

2. (a) Shown below is a cuboid.

 

Volume formula GCSE question 2

 

The dimensions of the cuboid are 1.2 m,80 cm 1.2 \ m, 80 \ cm and 55 cm. 55 \ cm.

Find the volume of the cuboid.

Give your answer in cubic metres.

 

(b) The cuboid is filled with water.

You are given that 1 m3=1000 litres. 1 \ m^3=1000 \ litres.

Calculate the volume of water needed to fill the cuboid.

Give your answer in litres.

 

(5 marks)

Show answer

(a)

 

80 cm=0.8 m 80 \ cm =0.8 \ m or 55 cm=0.55 m 55 \ cm = 0.55 \ m

(1)

1.2×0.8×0.55 1.2 \times 0.8 \times 0.55

(1)

0.528 m3 0.528 \ m^3

(1)

 

(b)

 

0.528×1000 0.528 \times 1000

(1)

528 litres 528 \ litres

(1)

3. (a) The volume of a cylinder is 1350 cm3. 1350 \ cm^3.

The radius of the cylinder is 6.8 cm. 6.8 \ cm.

Calculate the height of the cylinder.

Give your answer to 3 3 significant figures.

 

(b) The cylinder is made from pine wood.

Pine wood has a density of 373 kg/m3. 373 \ kg/m^{3}.

Given that Density=massvolume \text{Density}=\frac{\text{mass}}{\text{volume}} .

Calculate the mass of the cylinder.

Give your answer in kilograms.

 

(3 marks)

Show answer

(a)

 

1350=π×6.82×h 1350=\pi \times 6.8^{2} \times h

(1)

h=1350π×6.82 h=\frac{1350}{\pi \times 6.8^2}

(1)

9.29 cm 9.29 \ cm

(1)

 

(b)

 

1350 cm3=0.00135 m3 1350 \ cm^3=0.00135 \ m^3

(1)

373×0.00135 373 \times 0.00135

(1)

0.50355 kg 0.50355 \ kg

(1)

4. The diagram shows a hollow pipe. It is made from a cylinder with diameter 10 m 10 \ m and height 16 m, 16 \ m, with a cylindrical hole of diameter 8 m 8 \ m and height 16 m. 16 \ m.

 

Volume formula GCSE question 4

 

Work out the volume of the hollow pipe.

Leave your answer in terms of π. \pi. Include the units.

 

(5 marks)

Show answer

r1=5  r_1=5 \ or  r2=4  \ r_2=4 \

(1)

V1=π×52×16  V_1=\pi \times 5^2 \times 16 \ or  V2=π×42×16 \ V_2=\pi \times 4^2 \times 16

(1)

400π256π 400\pi-256\pi

(1)

144π 144\pi

(1)

m3 m^3

(1)

5. The volume of a sphere is 3050 mm3. 3050 \ mm^3.

Calculate the diameter of the sphere.

Give your answer correct to 3 3 significant figures.

 

(4 marks)

Show answer
3050=43×π×r3 3050=\frac{4}{3} \times \pi \times r^3

(1)

r=3×30504×π3 r=\sqrt[3]{\frac{3\times3050}{4\times \pi}}

(1)

d=2×8.99643 d=2\times8.99643…

(1)

18.0 cm 18.0 \ cm

(1)

Learning checklist

You have now learned how to:

  • Use a volume formula to calculate the volume of a 3D shape
  • Use a volume formula to find a missing dimension of a shape given its volume
  • Solve problems involving volume using a volume formula

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