Arc length and sector area GCSE Questions and Worked Answers

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Start with the meaning

What you need to know about arc length and sector area

Cut a circular piece of card into four equal slices through its centre. Each slice covers one quarter of the circle. Its curved edge is one quarter of the circle's boundary. The filled slice is called a sector; the curved part of its boundary is called an arc.

See the idea first

The angle tells you the fraction of a whole turn

The radius r runs from centre to edge. The angle between the two radii is the central angle, often written θ, pronounced ‘theta’. In degrees, a full circle has 360°. A 90° sector therefore uses 90/360 = 1/4 of both the whole circumference and the whole area.

The two straight edges are radii. The curved edge is the arc; the filled part is the sector.

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 curved length

What the problem asks: Find the arc length of a 90° sector of radius 6 cm.

How to solve it: The whole circumference is 2π × 6 = 12π cm. One quarter is 3π cm.

Find a sector area

What the problem asks: Find the area of the same 90° sector of radius 6 cm.

How to solve it: The whole area is π × 6² = 36π cm². One quarter is 9π cm². This is an area, not a length.

Find the whole boundary

What the problem asks: Find the perimeter of that sector.

How to solve it: Add the arc and both straight radii: 3π + 6 + 6 = 12 + 3π cm.

Recover a missing angle

What the problem asks: An arc is 4π cm long in a circle of radius 8 cm. Find its central angle.

How to solve it: The circumference is 16π cm, so the arc is one quarter of the circle. Its central angle is one quarter of 360°, or 90°.

A reliable routine

Take the same fraction of the whole circle

Use this when a region is bounded by two radii and one arc. The central angle determines what fraction of the circle is included. Arc length and area both use that fraction, but start with different whole-circle quantities.

Find radius r, halving any given diameter.
Calculate the sector fraction θ ÷ 360, with θ in degrees.
Multiply by 2πr for curved length or by πr² for area.
For perimeter add 2r; for reverse questions compare the given part with the corresponding whole circle.

Check: For a major sector, use the larger angle, 360° minus the minor angle. A chord is a straight join between two edge points; it is not an arc.

Fully worked

Arc length and sector area GCSE worked examples

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

Example 1

Arc length

3 marks

Question

A sector has radius 9 cm and central angle 80°. Find its arc length in terms of π.

\text{Fraction}=\frac{80}{360}=\frac29

L=\frac29(2\pi\cdot9)

=4\pi\text{ cm}

Tip: use circumference, not circle area.

Example 2

Sector area

3 marks

Question

Find the area of a 120° sector with radius 6 cm in terms of π.

The fraction is $120/360=1/3$ .

A=\frac13\pi(6)^2

=12\pi\text{ cm}^2

Example 3

Sector perimeter

3 marks

Question

A sector has radius 5 cm and angle 72°. Find its perimeter in terms of π.

L=\frac{72}{360}(2\pi\cdot5)=2\pi\text{ cm}

Add both radii.

P=2\pi+5+5=10+2\pi\text{ cm}

Example 4

Angle from arc

3 marks

Question

A circle has radius 7 cm. An arc has length $7\pi$ cm. Find the central angle.

Whole circumference: $2\pi(7)=14\pi$ cm.

\text{Fraction}=\frac{7\pi}{14\pi}=\frac12

\theta=\frac12(360^\circ)=180^\circ

Example 5

Major sector

3 marks

Question

A circle of radius 4 cm has a minor central angle of 90°. Find the area of the remaining major sector.

The larger angle is $360^\circ-90^\circ=270^\circ$ .

A=\frac{270}{360}\pi(4)^2

=\frac34(16\pi)=12\pi\text{ cm}^2

Use the sector actually requested.

Example 6

Find radius from area

4 marks

Question

A 60° sector has area $24\pi$ cm². Find its radius.

\frac{60}{360}\pi r^2=24\pi

Cancel π and multiply by 6.

r^2=144

r=12\text{ cm}

Use the positive square root for a radius.

10 original questions · total 27 marks

Arc length and sector area GCSE exam-style questions

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

Before you startAllow about 32 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

Quarter-circle arc

2 marks

Find the arc length of a 90° sector with radius 8 cm.

Show worked answer

L=\frac14(2\pi\cdot8)=4\pi\text{ cm}

2

Half-circle area

2 marks

Find the area of a 180° sector with radius 3 cm.

Show worked answer

A=\frac12\pi(3)^2=\frac{9\pi}{2}\text{ cm}^2

3

General arc

3 marks

Find the arc length for radius 12 cm and angle 150°.

Show worked answer

L=\frac{150}{360}(24\pi)

=\frac5{12}(24\pi)=10\pi\text{ cm}

4

General area

3 marks

Find the sector area for radius 10 cm and central angle 36°.

Show worked answer

A=\frac{36}{360}(100\pi)=10\pi\text{ cm}^2

5

Diameter given

3 marks

A sector has diameter 14 cm and angle 180°. Find its area.

Show worked answer

Radius is 7 cm.

A=\frac12\pi(7)^2=\frac{49\pi}{2}\text{ cm}^2

6

Perimeter

3 marks

Find the perimeter of a 90° sector of radius 6 cm.

Show worked answer

Arc: $\frac14(12\pi)=3\pi$ cm. Add 12 cm from the two radii:

P=12+3\pi\text{ cm}

7

Missing angle from area

3 marks

A sector has area $15\pi$ cm² in a circle of radius 5 cm. Find its angle.

Show worked answer

The whole area is $25\pi$ cm². The fraction is $15/25=3/5$ .

\theta=\frac35(360^\circ)=216^\circ

This is a major sector.

8

Radius from arc

3 marks

A 120° sector has arc length $8\pi$ cm. Find its radius.

Show worked answer

\frac13(2\pi r)=8\pi

Multiply both sides by 3.

2\pi r=24\pi

Divide both sides by π.

2r=24

r=12\text{ cm}

9

Rounded arc

3 marks

Find the arc length of a 100° sector of radius 7 cm to three significant figures.

Show worked answer

L=\frac{100}{360}(14\pi)

=\frac{35\pi}{9}=12.2173\ldots

Arc length is 12.2 cm to three significant figures. Keep π until the final rounding.

10

A false formula

2 marks

A student finds sector perimeter using θ/360 × (2πr + 2r). Explain the mistake.

Show worked answer

Only the curved circumference is scaled by the sector fraction. The two straight sides are full radii. The correct expression is θ/360 × 2πr + 2r.

Examiner-style feedback

Common arc length and sector area mistakes

Arc confused with perimeter

The sector has two straight radii as well as its arc.

Using the wrong angle

A major sector uses the larger central angle: 360° minus the minor angle, not the minor angle itself.

Squaring in the length formula

Arc length comes from 2πr; area comes from πr².

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Find the radius.
Use central angle ÷ 360.
Choose circumference or area.
Include radii only for perimeter.

Quick answers

Arc length and sector area FAQ

Are radians needed?

No. These GCSE questions use degrees and the fraction θ/360.

Can I keep π in an answer?

Yes when an exact answer or an answer in terms of π is requested. Otherwise round only at the end.

Build connected skills

What to revise next

Whole circles

Area and perimeter

Review radius, circumference and circle area.

Revise area and perimeter

Angles

Angle rules

Work with full turns and missing angles.

Revise angle rules

Content standards

Curriculum and rights review

G18 degree-based arc lengths, central angles, sectors and perimeters across tiers; no radians or calculus. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content