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Edexcel IGCSE Maths (4MA1) · Unit 5: Geometry

Edexcel IGCSE 4MA1 Volume and Surface Area: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

The volume of any prism is the area of its cross-section times its length, so a cylinder has volume $π r^{2} h$ . A cone or pyramid is a third of the matching prism, and a sphere has volume $\frac{4}{3} π r^{3}$ . Surface area is the total area of every face: a closed cylinder has two circles plus the curved surface $2 π r h$ . The 4MA1 Higher formulae sheet gives the prism, cylinder, cone and sphere formulas, but not the pyramid.

Unit 5: Geometry
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Worked examples
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Volume and Surface Area notes

Prisms and cylinders

Volume = area of cross-section × length. A triangular prism whose cross-section is a right-angled triangle with shorter sides 3 cm and 4 cm, and whose length is 10 cm, has volume $\frac{1}{2} \times 3 \times 4 \times 10 = 60$ cm³. A cylinder of radius 3 cm and height 10 cm has volume $90 π \approx 283$ cm³.

Surface area

Add the areas of all the faces; a sketch of the net helps you count them. The curved surface of a cylinder unrolls to a rectangle $2 π r$ by $h$ . A closed cylinder with $r = 3$ and $h = 10$ has surface area $2 π \times 9 + 2 π \times 3 \times 10 = 78 π \approx 245$ cm².

Cones and spheres

Cone: $V = \frac{1}{3} π r^{2} h$ and curved surface area $π r l$ , where $l$ is the slant height. Sphere: $V = \frac{4}{3} π r^{3}$ and surface area $4 π r^{2}$ . A cone with $r = 6$ and $h = 8$ has slant height $l = \sqrt{6^{2} + 8^{2}} = 10$ , volume $96 π$ and curved surface area $60 π$ .

Pyramids and frustums

Pyramid: $V = \frac{1}{3} \times base area \times h$ . A square-based pyramid with base 6 cm by 6 cm and height 5 cm has volume $\frac{1}{3} \times 36 \times 5 = 60$ cm³. A frustum is a cone or pyramid with its top cut off: its volume is the large cone minus the small one.

Units and working backwards

Volumes are in cubic units and areas in square units, and 1 m³ = 1 000 000 cm³. To find a missing length, substitute and solve: a cylinder of volume 500 cm³ and radius 5 cm has height $\frac{500}{25 π} \approx 6.37$ cm.

Key formulas

Prism; cylinder On the formula sheet: $V = cross-section \times length; V = π r^{2} h, curved A = 2 π r h$
Cone (slant $l$ ) On the formula sheet: $V = \frac{1}{3} π r^{2} h, curved A = π r l$
Sphere On the formula sheet: $V = \frac{4}{3} π r^{3}, A = 4 π r^{2}$
Pyramid Learn it: $V = \frac{1}{3} \times base area \times h$

On the formula sheet = printed on the formulae sheet in the 4MA1 Higher papers, so know where to find it. Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Volume and Surface Area worked examples

Worked example 1: Volume of a prism

A triangular prism has a cross-section with base 6 cm and height 4 cm, and length 15 cm. Find its volume.

Triangular prism: triangular cross-section with base 6 cm and height 4 cm, length 15 cm

Solution

Cross-section area: $\frac{1}{2} \times 6 \times 4 = 12$ .
Volume $= 12 \times 15 = 180$ cm³.

3 steps

Tip: Volume of a prism = area of cross-section × length.

Worked example 2: Surface area of a cylinder

Find the total surface area of a closed cylinder with radius 5 cm and height 8 cm.

Solution

Two circles: $2 π \times 25 = 50 π$ .
Curved surface: $2 π \times 5 \times 8 = 80 π$ .
Total $130 π = 408$ cm² (3 s.f.).

4 steps

Tip: The curved surface unrolls to a rectangle 2πr by h.

Worked example 3: Volume of a sphere

Find the volume of a sphere of radius 4.5 cm.

Solution

$V = \frac{4}{3} π r^{3} = \frac{4}{3} π \times 91.125$ .
$= 382$ cm³ (3 s.f.).

3 steps

Tip: Cube the radius before multiplying.

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Volume and Surface Area exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

A cube has edges of length 4 cm. Work out its total surface area.

Show the answer and mark scheme

M1 6 × 4²
A1 96 cm²

Answer: 96 cm²

Question 2 · easy · 2 marks

A cuboid measures 5 cm by 3 cm by 2 cm. Work out its total surface area.

Show the answer and mark scheme

M1 2 × (5×3 + 5×2 + 3×2) or at least two different face areas doubled
A1 62 cm²

Answer: 62 cm²

Question 3 · easy · 3 marks

A closed cylinder has radius 4 cm and height 10 cm. Work out its total surface area. Give your answer in terms of π.

Show the answer and mark scheme

M1 2 × π × 4² (= 32π)
M1 2 × π × 4 × 10 (= 80π)
A1 112π

Answer: $112 π$ cm²

Question 4 · medium · 3 marks

A triangular prism has a cross-section that is a right-angled triangle with sides 6 cm, 8 cm and 10 cm. The prism is 15 cm long. Work out its total surface area.

Full worked solution Free, no sign-in

M1 2 × ½ × 6 × 8 (= 48)
M1 15 × (6 + 8 + 10) (= 360)
A1 408 cm²

Answer: 408 cm²

Every question here has a full worked solution. See them all →

Question 5 · medium · 3 marks

A cylinder with no lid has radius 5 cm and height 12 cm. Work out its total outside surface area. Give your answer to 3 significant figures.

Show the answer and mark scheme

M1 π × 5² (one end only)
M1 2 × π × 5 × 12
A1 456 (455.5…)

Answer: 456 cm²

Question 6 · medium · 2 marks

The curved surface area of a cylinder is $150 π$ cm². The radius is 5 cm. Find the height of the cylinder.

Show the answer and mark scheme

M1 2π × 5 × h = 150π
A1 15 cm

Answer: 15 cm

Question 7 · hard · 3 marks

A cuboid has a square base of side 6 cm and a volume of 360 cm³. Work out its total surface area.

Show the answer and mark scheme

M1 height 360 ÷ 36 = 10
M1 2 × 36 + 4 × 6 × their 10
A1 312 cm²

Answer: 312 cm²

Question 8 · hard · 4 marks

A prism is 20 cm long. Its cross-section is an isosceles trapezium with parallel sides 10 cm and 4 cm, perpendicular height 4 cm and sloping sides 5 cm. Work out the total surface area of the prism.

Show the answer and mark scheme

M1 trapezium area ½ × (10 + 4) × 4 (= 28)
M1 2 × their 28
M1 20 × (10 + 4 + 5 + 5) (= 480)
A1 536 cm²

Answer: 536 cm²

Common volume and surface area mistakes

Using the diameter as the radius. Halve the diameter first. Putting the diameter into $π r^{2}$ makes the area four times too big.
Missing faces. A closed cylinder has two circular ends as well as the curved surface; an open one, like a cup, has one end.
Slant height or vertical height. A cone's volume uses the vertical height $h$ ; its curved surface uses the slant height $l$ . Pythagoras links them: $l^{2} = r^{2} + h^{2}$ .
Mixing units. Convert every length to the same unit before you calculate. Converting cm³ to m³ divides by 1 000 000, not by 100.

Where students lose marks across 4MA1, from the examiner reports →

Volume and Surface Area FAQ

Which volume formulas are on the 4MA1 formula sheet?

The 4MA1 Higher formulae sheet gives the volume of a prism, the volume and curved surface area of a cylinder and of a cone, and the volume and surface area of a sphere. The pyramid formula, V = ⅓ × base area × height, is not given, so learn it.

How do I find the surface area of a cylinder?

Add the two circular ends, 2πr², to the curved surface, 2πrh. For an open cylinder, such as a tin with no lid, include only one end.

Can I use a calculator for volume and surface area in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Volume and Surface Area revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 surface area and volume questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Volume and Surface Area: exam questions and answers, notes and worked examples

Try it first

Volume and Surface Area notes

Prisms and cylinders

Surface area

Cones and spheres

Pyramids and frustums

Units and working backwards

Key formulas

Volume and Surface Area worked examples

Worked example 1: Volume of a prism

Solution

Worked example 2: Surface area of a cylinder

Solution

Worked example 3: Volume of a sphere

Solution

Volume and Surface Area exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 2 marks

Question 3 · easy · 3 marks

Question 4 · medium · 3 marks

Full worked solution Free, no sign-in

Question 5 · medium · 3 marks

Question 6 · medium · 2 marks

Question 7 · hard · 3 marks

Question 8 · hard · 4 marks

Common volume and surface area mistakes

Volume and Surface Area FAQ

Next steps

Practise volume and surface area with your progress saved

More Unit 5 topics: Geometry