Learn9709Paper 1 · Pure Mathematics 1Circular measureArc length and sector area

Circular measure · Subtopic 2 of 3

Arc Length and Sector Area

Why the formulae $s=r\theta$ and $A=\tfrac12r^2\theta$ work, how to use them for perimeters, and how to work backwards from a length or an area to the radius or angle.

Cambridge 9709 Paper 1Edexcel YMA01 WMA11Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1AQA 7357 Paper 2≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find arc lengths with $s=r\theta$ and sector areas with $A=\tfrac12r^2\theta$ .
Find the perimeter of a sector, remembering both radii.
Rearrange the formulae to find an unknown radius or angle.
Convert from degrees first when an angle is given in degrees.

Pause and think

A pizza of radius $15$ cm is cut into $8$ equal slices. What fraction of the whole pizza is one slice, and what angle (in radians) is at its tip?

Section 2 of 14

Core idea

Start hereHow much pizza is in a slice?

A pizza of radius $15$ cm is cut into a slice with a $0.8$ radian angle. How long is the crust edge, and how much pizza is it? The slice is a fraction of the whole: in radians that fraction is $\frac{\theta}{2\pi}$ , and the formulas become beautifully short: crust $15\times0.8=12$ cm, area $\frac12\times15^2\times0.8=90$ cm².

The big idea

With $\theta$ in radians: arc length $s=r\theta$ , sector area $A=\tfrac12r^2\theta$ . Both come from taking the fraction $\frac{\theta}{2\pi}$ of the whole circle.

1. Deriving the formulas

A sector growing: arc and area

θ (radians) = 0.30

\theta

grows to

2\pi

, the arc becomes the whole circumference

6\pi

and the area becomes

\pi r^2=9\pi

Arc length and sector area

$\text{fraction of circle}=\frac{\theta}{2\pi}$
A full turn is $2\pi$ radians.
$s=\frac{\theta}{2\pi}\times2\pi r=r\theta$
That fraction of the circumference.
$A=\frac{\theta}{2\pi}\times\pi r^2=\tfrac12r^2\theta$
That fraction of the area.

These need

\theta

in radians. (In degrees:

s=\frac{\theta}{360}\times2\pi r

2. Using them

Worked exampleSector $r=6$ cm, $\theta=\tfrac{2\pi}3$

Find the arc length, area and perimeter.

Goal: Exact values and decimals.
Strategy: Substitute; perimeter = arc + two radii.

$s=6\times\tfrac{2\pi}3=4\pi\approx12.6\text{ cm}$
$s=r\theta$ .
$A=\tfrac12\times36\times\tfrac{2\pi}3=12\pi\approx37.7\text{ cm}^2$
$A=\frac12r^2\theta$ .
$P=4\pi+12\approx24.6\text{ cm}$
Don't forget the two straight edges.

Answer

4\pi

cm,

12\pi

cm²,

4\pi+12

cm.

✓ Check

\frac{2\pi}3

is a third of a turn: area

\frac13\times36\pi=12\pi

✓.

Worked examplePerimeter given, find the radius

A sector has angle $0.8$ rad and perimeter $28$ cm. Find $r$ and the area.

Goal: $r$ , then $A$ .
Strategy: Write perimeter in terms of $r$ , solve.

$r\theta+2r=28\Rightarrow0.8r+2r=28$
Arc + two radii.
$2.8r=28\Rightarrow r=10$
Factor $r$ .
$A=\tfrac12(100)(0.8)=40\text{ cm}^2$
Area formula.

Answer

r=10

cm; area

40

cm².

✓ Check

8+20=28

✓.

Common misconceptionMixing degrees and radians

s=r\theta

with

\theta=60

(degrees) gives nonsense. Convert first:

60^\circ=\frac\pi3

Pause and think

A pendulum of length 80 cm swings through 0.3 rad. How far does its tip travel?

Summary

$s=r\theta$ , $A=\frac12r^2\theta$ (radians).
Perimeter $=r\theta+2r$ .

Arc: Part of the circumference.
Sector: Region between two radii and an arc.

HistoryMeasuring the Earth with an arc. Around 240 BC, Eratosthenes noticed that at noon on midsummer the Sun was overhead in Syene but about

7.2^\circ

from overhead in Alexandria, further north.

7.2^\circ

\tfrac1{50}

of a full turn, so the distance between the cities had to be

\tfrac1{50}

of the Earth's circumference. His estimate was remarkably close to the true value.

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