Learn8300Papers 1–3 · Higher tierGeometry and measuresCircles, arcs and sectors

Geometry and measures · Subtopic 7 of 17

Circles, Arcs and Sectors

Circumference $2\pi r$ and area $\pi r^2$ . A sector is a fraction $\dfrac{\theta}{360}$ of the circle, so its arc and area are that fraction of the circumference and area. Its perimeter is the arc plus two radii.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find the circumference and area of a circle, including from the diameter.
Find arc length and sector area using the fraction $\dfrac{\theta}{360}$ .
Find the perimeter of a sector or semicircle.
Work backwards to find a radius or an angle.

Pause and think

A pizza is cut into $8$ equal slices. What fraction of the full circle is one slice, and what angle is it at the centre?

Section 2 of 14

Core idea

Start hereHow far does a wheel roll?

One turn of a bicycle wheel moves the bike forward by exactly its circumference. Pi, $3.14159\ldots$ , is the number of diameters that fit around any circle.

The big idea

$C=2\pi r=\pi d$ ; $A=\pi r^2$ . Sector of angle $\theta$ : fraction $\frac\theta{360}$ of the circle.

1. Why πr²

Unrolling a circle into a near-rectangle shows A = πr²

Cut the circle into thin slices and rearrange: you get a shape close to a rectangle,

\pi r

long and

r

high. Area

=\pi r\times r=\pi r^2

Unrolling the circle

$\text{cut into many thin sectors}$
Like pizza slices.
$\text{rearrange: width}\approx\tfrac12(2\pi r)=\pi r,\ \text{height}=r$
Half the slices point up, half down.
$A=\pi r\times r=\pi r^2$
Becomes a rectangle as slices get thinner.

2. Sectors and arcs

A sector with angle θ

θ (degrees) = 30

Sector area

=\dfrac{\theta}{360}\times\pi r^2

; arc length

=\dfrac{\theta}{360}\times2\pi r

\text{arc}=\frac\theta{360}\times2\pi r\qquad\text{sector}=\frac\theta{360}\times\pi r^2

$\theta$: angle at the centre (degrees).

Worked example $r=6$ , $\theta=50^\circ$

Arc, area, perimeter.

Goal: Three values.
Strategy: Fraction $\frac{50}{360}$ .

$\tfrac{50}{360}\times12\pi=5.24$
Arc.
$\tfrac{50}{360}\times36\pi=15.7$
Area.
$5.24+6+6=17.24$
Perimeter: arc + two radii.

Answer

5.24

cm;

15.7

cm²;

17.24

cm.

✓ Check

50^\circ

is about

\frac17

of a circle;

\frac17\times113=16

✓.

Worked exampleWorking backwards

Circumference 50 cm.

Goal: Radius and area.
Strategy: Rearrange $C=2\pi r$ .

$r=\frac{50}{2\pi}=7.96$
Divide.
$A=\pi(7.96)^2=199$
Area.

Answer

7.96

cm;

199

cm².

✓ Check

2\pi(7.96)=50.0

✓.

Common misconceptionDiameter in πr² Halve the diameter first.

Pause and think

A wheel of radius 30 cm turns 100 times. Distance?

Summary

$2\pi r$ , $\pi r^2$ .
Sector fraction $\theta/360$ .

Sector: Slice of a circle.
Arc: Part of the circumference.

HistoryArchimedes squeezes π. Around

250

BC Archimedes trapped

\pi

between the perimeters of

96

-sided polygons drawn inside and outside a circle, proving

3\tfrac{10}{71}<\pi<3\tfrac17

. That is where the school value

\tfrac{22}7

comes from.

Fun factPi Day.

\pi

is celebrated on 14 March (

3/14

in the American date format), which is also Albert Einstein's birthday. Computers have now calculated

\pi

to over a hundred trillion digits, though NASA needs only about

15

to steer spacecraft.

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