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Volume and Surface Area

Volumes and surface areas of cuboids, prisms, cylinders, cones and spheres.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Calculate the volume and surface area of cuboids and prisms, using the area of the cross-section.
  2. 2Calculate the volume and surface area of cylinders, with answers in terms of \(\pi\) where asked.
  3. 3Calculate the volume and surface area of cones, spheres and pyramids (Higher tier).
  4. 4Use volume to solve problems about capacity and density.

Space inside and skin outside

Volume is the amount of space inside a 3D shape, measured in cubic units such as cm\(^3\). Surface area is the total area of all the faces, which is measured in square units, because it is the area of the "skin" of the shape. Examiners ask for both, and the units tell you which is which. Most volume questions use a simple idea: find the area of one end of the shape, then multiply by how far it goes.

Cuboids

A cuboid is a box shape with three different dimensions, length \(l\), width \(w\) and height \(h\).

  • Volume

    \(V = l \times w \times h\).

  • Surface area

    A cuboid has six faces in three matching pairs: \(2lw + 2lh + 2wh\).

  • A cube

    A cube has all its edges equal, so \(V = a^3\) and the surface area is \(6a^2\).

  • Always write the units

    Volume is in cm\(^3\), and surface area is in cm\(^2\).

Cuboid volume and surface area

A cuboid measures 8 cm by 5 cm by 3 cm. Work out its volume and its total surface area.

Show the solutionHide the solution
  1. 1 Volume \(8 \times 5 \times 3 = 120\) cm\(^3\).
  2. 2 Find the three pairs of faces \(8 \times 5 = 40\), \(8 \times 3 = 24\) and \(5 \times 3 = 15\).
  3. 3 Add and double \(40 + 24 + 15 = 79\), so \(2 \times 79 = 158\).
  4. 4 Units 158 cm\(^2\).

AnswerVolume 120 cm\(^3\), surface area 158 cm\(^2\)

Prisms

A prism has the same cross-section all the way through. It could be a triangle, a trapezium, an L-shape or any other flat shape.

  • Volume

    Volume \(=\) area of cross-section \(\times\) length. This one rule covers every prism.

  • Find the cross-section first

    Work out its area using the formulae from the last lesson, then multiply.

  • Surface area

    Find the area of every face, including both ends, and add them. A net helps you not to miss one.

  • The length is the distance along the prism

    It is not always the longest dimension in the picture.

Volume of a triangular prism

The cross-section of a prism is a right-angled triangle with base 6 cm and height 4 cm. The prism is 15 cm long. Work out its volume.

Show the solutionHide the solution
  1. 1 Area of the triangle \(\dfrac{1}{2} \times 6 \times 4 = 12\) cm\(^2\).
  2. 2 Multiply by the length \(12 \times 15 = 180\).
  3. 3 Units 180 cm\(^3\).

Answer180 cm\(^3\)

Cylinders

A cylinder is a prism with a circle as its cross-section.

  • Volume

    \(V = \pi r^2 h\), the area of the circle times the height.

  • Curved surface area

    \(2\pi r h\). The curved surface opens out into a rectangle of height \(h\) and length equal to the circumference.

  • Total surface area

    The curved surface plus the two circles: \(2\pi r h + 2\pi r^2\).

  • Take care with the radius

    If a diameter is given, halve it first.

Volume and surface area of a cylinder

A cylinder has radius 3 cm and height 10 cm. Work out its volume and its total surface area. Give your answers in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 Volume \(\pi r^2 h = \pi \times 3^2 \times 10 = 90\pi\) cm\(^3\).
  2. 2 Curved surface \(2\pi r h = 2 \times \pi \times 3 \times 10 = 60\pi\) cm\(^2\).
  3. 3 The two circles \(2\pi r^2 = 2 \times \pi \times 9 = 18\pi\) cm\(^2\).
  4. 4 Total \(60\pi + 18\pi = 78\pi\) cm\(^2\).

AnswerVolume \(90\pi\) cm\(^3\), surface area \(78\pi\) cm\(^2\)

Cones, spheres and pyramids (Higher tier)

These formulae appear on the Higher paper, and you should know how to use them even when they are given on the formula page.

  • Cone volume

    \(V = \dfrac{1}{3}\pi r^2 h\), using the vertical height \(h\).

  • Cone curved surface

    \(\pi r l\), where \(l\) is the slant height, not the vertical height.

  • Sphere

    Volume \(\dfrac{4}{3}\pi r^3\) and surface area \(4\pi r^2\).

  • Pyramid

    Volume \(\dfrac{1}{3} \times\) area of base \(\times\) vertical height.

  • Hemisphere

    Half a sphere. Its volume is \(\dfrac{2}{3}\pi r^3\), and its curved surface is \(2\pi r^2\), plus \(\pi r^2\) if the flat circle is included.

Volume of a sphere

(Higher tier) A sphere has radius 3 cm. Work out its volume, leaving your answer in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 Use the formula \(V = \dfrac{4}{3}\pi r^3\).
  2. 2 Cube the radius \(3^3 = 27\).
  3. 3 Substitute \(\dfrac{4}{3} \times \pi \times 27\).
  4. 4 Calculate \(\dfrac{4 \times 27}{3} = 36\), so the volume is \(36\pi\) cm\(^3\).

Answer\(36\pi\) cm\(^3\)

Volume of a cone

(Higher tier) A cone has radius 3 cm and vertical height 4 cm. Work out its volume in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 Use the formula \(V = \dfrac{1}{3}\pi r^2 h\).
  2. 2 Substitute \(\dfrac{1}{3} \times \pi \times 9 \times 4\).
  3. 3 Calculate \(\dfrac{36}{3} = 12\), so the volume is \(12\pi\) cm\(^3\).
  4. 4 Note The slant height of this cone is 5 cm, because \(3^2 + 4^2 = 5^2\), but it is not needed for the volume.

Answer\(12\pi\) cm\(^3\)

Problems using volume

Volume links to other topics, and these are the classic ways it is tested.

  • Capacity

    1 cm\(^3\) holds 1 ml, so a volume in cm\(^3\) can be changed to millilitres, and 1000 cm\(^3\) is 1 litre.

  • Density

    Mass \(=\) density \(\times\) volume, from the previous chapter.

  • Filling and pouring

    A tank has volume \(l \times w \times h\), and the time to fill it is volume divided by the rate.

  • Cutting a shape

    Subtract the volume that is removed.

What the Edexcel exam gives you

Edexcel prints a formulae page in the question paper, so some formulae are given to you.

  • Given on the formulae page

    The volume of a prism: area of cross-section \(\times\) length.

  • Given in the question

    The cone, sphere and pyramid formulae are given in the question when they are needed (Higher tier).

  • Not given

    The volume of a cuboid, the volume of a cylinder \(\pi r^2 h\), and the surface area of a cylinder are not given, so learn them.

Test yourself

  1. 1

    What is the volume of a prism?

    Show answerHide answer

    Area of cross-section \(\times\) length.

  2. 2

    What is the volume of a cylinder?

    Show answerHide answer

    \(\pi r^2 h\).

  3. 3

    What is the curved surface area of a cylinder?

    Show answerHide answer

    \(2\pi r h\).

  4. 4

    How many millilitres are in 1 cm\(^3\)?

    Show answerHide answer

    1 ml.

  5. 5

    What is the volume of a sphere (Higher tier)?

    Show answerHide answer

    \(\dfrac{4}{3}\pi r^3\).

Exam technique: volume and surface area

Marks are given for each correct area or volume, so show them separately.

  • Write the cross-section area as its own line

    It is worth a method mark even if the volume is wrong.

  • Check the units

    cm\(^3\) for volume, cm\(^2\) for surface area.

  • Count the faces

    A prism has two ends and the rectangles along its sides.

  • Leave \(\pi\) in the answer

    Only change \(\pi\) to a number if the question asks for a decimal.

Summary and exam focus

  • Volume of a cuboid is \(l \times w \times h\), and of any prism it is the cross-section area times the length.
  • A cylinder has volume \(\pi r^2 h\) and curved surface area \(2\pi r h\).
  • Total surface area is the area of every face added together.
  • Cones, spheres and pyramids appear at Higher tier, with \(\dfrac{1}{3}\) used for cones and pyramids.
  • 1 cm\(^3\) is 1 ml, and 1000 cm\(^3\) is 1 litre.

Exam focus

A cylinder has radius 4 cm and height 10 cm. Work out the volume of the cylinder. Give your answer in terms of \(\pi\). (2 marks) (2 marks)

Write \(\pi \times 4^2 \times 10\) first, then square the 4 to get 16 and multiply by 10. The answer is \(160\pi\) cm\(^3\). Do not square the 4 as \(4 \times 2 = 8\), which is the most common slip.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Volume
The amount of space inside a 3D shape, measured in cubic units.
Surface area
The total area of all the faces of a 3D shape.
Cuboid
A 3D shape with six rectangular faces.
Prism
A 3D shape with the same cross-section along its whole length.
Cross-section
The shape you see when you cut through a solid at a right angle to its length.
Cylinder
A prism with a circular cross-section.
Cone
A 3D shape with a circular base that rises to a point.
Sphere
A perfectly round 3D shape, like a ball.
Net
A flat pattern that folds to make a 3D shape.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 5 marks Core

The diagram shows a prism. The cross-section of the prism is a right-angled triangle. (a) Work out the volume of the prism. (2 marks) (b) Work out the total surface area of the prism. (3 marks)

A triangular prism whose cross-section is a 3, 4, 5 right-angled triangle and whose length is 10 cm.

Mark scheme — 5 marks available

  • (a) \(\dfrac{1}{2} \times 4 \times 3 = 6\) — M1
  • (a) 60 cm\(^3\) — A1
  • (b) \(40\), \(30\) and \(50\) found, or \(12 \times 10\) — M1
  • (b) \(12\) for the triangles added to their rectangles — M1
  • (b) 132 cm\(^2\) — A1

Model answer

(a) The area of the triangle is \(\dfrac{1}{2} \times 4 \times 3 = 6\) cm\(^2\), so the volume is \(6 \times 10 = 60\) cm\(^3\). (b) The two triangles have area \(2 \times 6 = 12\). The three rectangles have areas \(4 \times 10 = 40\), \(3 \times 10 = 30\) and \(5 \times 10 = 50\), which total 120. The total surface area is \(12 + 120 = 132\) cm\(^2\).

2. Exam question Work out 2 marks Easier

A cuboid has a volume of 240 cm\(^3\). Its length is 10 cm and its width is 6 cm. Work out its height.

Mark scheme — 2 marks available

  • \(240 \div (10 \times 6)\) or \(60h = 240\) — M1
  • 4 cm — A1

Model answer

\(10 \times 6 \times h = 240\), so \(60h = 240\) and \(h = 4\) cm.

3. Exam question Work out 4 marks Core

A cylinder has radius 5 cm and height 8 cm. Give your answers in terms of \(\pi\). (a) Work out the volume of the cylinder. (2 marks) (b) Work out the curved surface area of the cylinder. (2 marks)

Mark scheme — 4 marks available

  • (a) \(\pi \times 5^2 \times 8\) — M1
  • (a) \(200\pi\) — A1
  • (b) \(2 \times \pi \times 5 \times 8\) — M1
  • (b) \(80\pi\) — A1

Model answer

(a) \(\pi \times 5^2 \times 8 = 200\pi\) cm\(^3\). (b) \(2 \times \pi \times 5 \times 8 = 80\pi\) cm\(^2\).

4. Exam question Work out 3 marks Core

A tank is in the shape of a cuboid measuring 80 cm by 50 cm by 40 cm. The tank is full of water. Work out the volume of water in the tank in litres.

Mark scheme — 3 marks available

  • \(80 \times 50 \times 40\) — M1
  • \(160\,000\) cm\(^3\) — A1
  • 160 litres — A1

Model answer

\(80 \times 50 \times 40 = 160\,000\) cm\(^3\). Since \(1000\text{ cm}^3 = 1\) litre, the volume is \(160\) litres.

5. Exam question Show that 3 marks Stretch

A cylinder has radius \(x\) cm and height \(2x\) cm. Show that the volume of the cylinder is \(2\pi x^3\) cm\(^3\).

Mark scheme — 3 marks available

  • \(\pi \times x^2 \times h\) with \(h = 2x\) substituted — M1
  • \(\pi x^2 \times 2x\) — M1
  • \(2\pi x^3\) with a conclusion — C1

Model answer

Volume \(= \pi r^2 h = \pi \times x^2 \times 2x = 2\pi x^3\), as required.

6. Exam question Work out 4 marks Stretch

Give your answers in terms of \(\pi\). (a) A sphere has radius 6 cm. Work out the volume of the sphere. (2 marks) (b) Work out the curved surface area of a hemisphere of radius 6 cm. (2 marks)

Mark scheme — 4 marks available

  • (a) \(\dfrac{4}{3} \times \pi \times 216\) — M1
  • (a) \(288\pi\) — A1
  • (b) \(\dfrac{1}{2} \times 4\pi \times 36\) or \(2\pi r^2\) — M1
  • (b) \(72\pi\) — A1

Model answer

(a) \(\dfrac{4}{3} \times \pi \times 6^3 = \dfrac{4}{3} \times 216\pi = 288\pi\) cm\(^3\). (b) The surface area of a sphere is \(4\pi r^2 = 144\pi\), so half of it is \(72\pi\) cm\(^2\).

7. Multiple choice 1 mark Easier

A cuboid measures 8 cm by 5 cm by 3 cm. What is its volume?

  1. A 16 cm\(^3\)
  2. B 158 cm\(^3\)
  3. C 240 cm\(^3\)
  4. D 120 cm\(^3\) Correct

Why: \(8 \times 5 \times 3 = 120\) cm\(^3\).

8. Multiple choice 1 mark Easier

The cross-section of a prism has area 12 cm\(^2\). The prism is 15 cm long. What is its volume?

  1. A 27 cm\(^3\)
  2. B 90 cm\(^3\)
  3. C 180 cm\(^3\) Correct
  4. D 12 cm\(^3\)

Why: Volume \(=\) area of cross-section \(\times\) length \(= 12 \times 15 = 180\).

9. Multiple choice 1 mark Core

A cylinder has radius 3 cm and height 10 cm. What is its volume, in terms of \(\pi\)?

  1. A \(30\pi\) cm\(^3\)
  2. B \(90\pi\) cm\(^3\) Correct
  3. C \(60\pi\) cm\(^3\)
  4. D \(900\pi\) cm\(^3\)

Why: \(\pi r^2 h = \pi \times 9 \times 10 = 90\pi\).

10. Multiple choice 1 mark Core

A cylinder has radius 3 cm and height 10 cm. What is its curved surface area, in terms of \(\pi\)?

  1. A \(60\pi\) cm\(^2\) Correct
  2. B \(30\pi\) cm\(^2\)
  3. C \(90\pi\) cm\(^2\)
  4. D \(180\pi\) cm\(^2\)

Why: \(2\pi r h = 2 \times \pi \times 3 \times 10 = 60\pi\).

11. Multiple choice 1 mark Easier

What is the total surface area of a cube with side 4 cm?

  1. A 64 cm\(^2\)
  2. B 16 cm\(^2\)
  3. C 24 cm\(^2\)
  4. D 96 cm\(^2\) Correct

Why: A cube has 6 faces, each of area \(4 \times 4 = 16\). So \(6 \times 16 = 96\).

12. Multiple choice 1 mark Core

A tank holds 2500 cm\(^3\) of water. How many litres is this?

  1. A 25 litres
  2. B 0.25 litres
  3. C 2.5 litres Correct
  4. D 250 litres

Why: \(1000\text{ cm}^3 = 1\) litre, so \(2500 \div 1000 = 2.5\).

13. Multiple choice 1 mark Core

A cylinder has radius 2 cm and height 7 cm. What is its volume, in terms of \(\pi\)?

  1. A \(14\pi\) cm\(^3\)
  2. B \(28\pi\) cm\(^3\) Correct
  3. C \(56\pi\) cm\(^3\)
  4. D \(98\pi\) cm\(^3\)

Why: \(\pi \times 2^2 \times 7 = 28\pi\). Using the diameter instead of the radius would give \(98\pi\).

14. Multiple choice 1 mark Stretch

A sphere has radius 3 cm. What is its volume, in terms of \(\pi\)?

  1. A \(36\pi\) cm\(^3\) Correct
  2. B \(12\pi\) cm\(^3\)
  3. C \(27\pi\) cm\(^3\)
  4. D \(108\pi\) cm\(^3\)

Why: \(\dfrac{4}{3}\pi r^3 = \dfrac{4}{3} \times \pi \times 27 = 36\pi\).

15. Multiple choice 1 mark Stretch

A cone has radius 3 cm and vertical height 4 cm. What is its volume, in terms of \(\pi\)?

  1. A \(36\pi\) cm\(^3\)
  2. B \(15\pi\) cm\(^3\)
  3. C \(9\pi\) cm\(^3\)
  4. D \(12\pi\) cm\(^3\) Correct

Why: \(\dfrac{1}{3}\pi r^2 h = \dfrac{1}{3} \times \pi \times 9 \times 4 = 12\pi\). Using the slant height 5 gives \(15\pi\), which is wrong.