Surface area GCSE Questions and Worked Answers

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Start with the meaning

What you need to know about surface area

To wrap a box exactly, with no overlap, imagine covering every outside face with paper. The total paper needed is its surface area. This measures the outside covering, not the space inside the box, and is measured in square units such as cm².

See the idea first

Unfold the faces so none is missed

A net is a flat arrangement of the faces that can fold into the solid. A cube has six equal square faces, so a cube with side a has surface area 6a². A cuboid has three pairs of equal rectangles instead.

A cube has six square faces. This net shows all six without overlap, including faces hidden in a perspective drawing.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find a closed box's surface area

What the problem asks: A cuboid measures 5 cm by 3 cm by 2 cm. Find its total surface area.

How to solve it: Face areas are 15, 10 and 6 cm², with two of each. Add 2 × (15 + 10 + 6) = 62 cm².

Leave out a missing face

What the problem asks: The same box has no lid. Find the area of material.

How to solve it: The missing 5 cm by 3 cm lid has area 15 cm². Subtract it from the closed-box total: 62 − 15 = 47 cm².

Include a curved surface

What the problem asks: Find the surface area of a closed cylinder with radius 3 cm and height 5 cm.

How to solve it: The curved side unrolls into a rectangle of width 2πr and height h: 30π cm². Add two circular ends, 18π cm², giving 48π cm².

A reliable routine

Add the surfaces that are actually exposed

This works when a solid's boundary can be separated into non-overlapping faces or known curved surfaces. A net makes the counting visible. For joined solids, a touching face is inside, not part of the exposed boundary.

List faces, including hidden ones, and mark any missing or joined faces.
Calculate the area of each different face and its number of copies.
For a cylinder, use curved area 2πrh; for a cone, curved area πrl with slant height l.
Add only the required surfaces and state square units. A sphere uses its own formula, 4πr².

Check: Cone slant height follows the sloping surface, while perpendicular height goes straight from apex to base. They are different lengths.

Fully worked

Surface area GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Cube

2 marks

Question

Find the surface area of a cube of side 5 cm.

One square face has area $5^2=25$ cm².

S=6(25)=150\text{ cm}^2

All six faces count.

Example 2

Cuboid

3 marks

Question

Find the surface area of a closed cuboid measuring 7 cm by 4 cm by 3 cm.

The three face pairs have areas 28, 21 and 12 cm² per face.

S=2(28+21+12)

=2(61)

=122\text{ cm}^2

Tip: counting only visible faces misses the opposite ones.

Example 3

Open box

3 marks

Question

A thin-walled rectangular box is 8 cm long, 5 cm wide and 4 cm high, with no lid. Find its material area, ignoring thickness.

Base: $8(5)=40$ . Two long walls: $2(8)(4)=64$ . Two short walls: $2(5)(4)=40$ .

S=40+64+40=144\text{ cm}^2

There is no top face.

Example 4

Closed cylinder

3 marks

Question

A closed cylinder has radius 3 cm and height 5 cm. Find its total surface area in terms of π.

Unroll the curved surface into a rectangle. Its width is the distance around the circular rim, 2πr. Add two circles for a closed cylinder.

Curved side: $2\pi(3)(5)=30\pi$ . Two ends: $2\pi(3)^2=18\pi$ .

S=30\pi+18\pi=48\pi\text{ cm}^2

The net explains both contributions.

Example 5

Cone with its base

3 marks

Question

A cone has radius 4 cm and slant height 9 cm. Use curved area $\pi rl$ to find its total surface area, including the base.

S_{\rm curved}=\pi(4)(9)=36\pi

S_{\rm base}=\pi(4)^2=16\pi

S=52\pi\text{ cm}^2

The given 9 cm is slant height, which belongs in the curved-area formula.

Example 6

Joined cubes

Harder4 marks

Question

Two cubes, each of side 3 cm, are joined along one complete face. Find the exposed surface area.

Separate cubes have $2\times6\times3^2=108$ cm² of surface. The two faces at the join are now internal; each is 9 cm².

S=108-2(9)=90\text{ cm}^2

Remove both touching faces, not just one.

10 original questions · total 28 marks

Surface area GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 33 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

Cube

2 marks

Find the total surface area of a cube of side 4 cm.

Show worked answer

S=6(4^2)=6(16)=96\text{ cm}^2

2

Cuboid

3 marks

Find the surface area of a 6 cm by 5 cm by 2 cm closed cuboid.

Show worked answer

S=2(6\cdot5+6\cdot2+5\cdot2)

=2(30+12+10)=104\text{ cm}^2

3

No lid

3 marks

A cube-shaped box of side 6 cm has no lid. Ignore material thickness. Find material area.

Show worked answer

Five square faces remain.

S=5(6^2)=180\text{ cm}^2

4

Cylinder's curved side

2 marks

Find only the curved surface area of a cylinder with radius 2 cm and height 7 cm.

Show worked answer

S=2\pi rh=2\pi(2)(7)=28\pi\text{ cm}^2

Do not add the ends when only the curved side is requested.

5

Closed cylinder

3 marks

A closed cylinder has diameter 10 cm and height 6 cm. Find total surface area in terms of π.

Show worked answer

Radius is 5 cm.

S=2\pi(5)(6)+2\pi(5)^2

=60\pi+50\pi=110\pi\text{ cm}^2

6

Sphere

2 marks

Find a sphere's surface area for radius 7 cm, using $S=4\pi r^2$ .

Show worked answer

S=4\pi(7)^2=196\pi\text{ cm}^2

7

Closed hemisphere

3 marks

A hemisphere has radius 3 cm. Use sphere surface area $4\pi r^2$ to find its total surface area, including the flat base.

Show worked answer

Curved half-sphere: $2\pi(3)^2=18\pi$ . Flat circular base: $9\pi$ .

S=27\pi\text{ cm}^2

Half the sphere area alone misses the base.

8

Triangular prism

Harder4 marks

A right triangular prism has a 3–4–5 cm right-triangle cross-section and length 10 cm. Find total surface area.

Show worked answer

Each triangular end: $\frac12(3)(4)=6$ cm². Rectangular sides: $3(10)+4(10)+5(10)=120$ cm².

S=2(6)+120=132\text{ cm}^2

9

Missing edge

3 marks

A cube has total surface area 216 cm². Find its edge length.

Show worked answer

6a^2=216

a^2=36

a=6\text{ cm}

Choose the positive root because an edge length is positive.

10

Paint coverage

3 marks

A surface area of 42 m² needs two coats of paint. Each litre covers 6 m² for one coat. Find the litres needed, ignoring waste.

Show worked answer

Two coats require coverage of $2(42)=84$ m².

84/6=14\text{ litres}

Coverage is area per litre, not volume of the solid.

Examiner-style feedback

Common surface area mistakes

Multiplying three dimensions

That finds a cuboid's volume. Surface area adds face areas.

Missing bases

A closed cylinder has two ends; a closed hemisphere has one flat circular base.

Counting joined faces

Internal faces are not exposed and must be removed from the surface total.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

List exposed surfaces.
Find every face area.
Count copies without overlap.
Use square units.

Quick answers

Surface area FAQ

Why is cylinder curved area 2πrh?

Unrolled, its side is a rectangle whose width is the circular circumference 2πr and whose height is h.

Is a net required in an exam?

Not always, but a quick sketch or face list can prevent missing or double-counting faces.

Build connected skills

What to revise next

Flat-shape formulas

Area and perimeter

Calculate the area of each face.

Revise area and perimeter

Inside space

Volume

Distinguish filling a solid from wrapping it.

Revise volume

Content standards

Curriculum and rights review

G17 surface area across tiers, with formulas given for less familiar curved solids and harder composite examples labelled locally. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content