LearnYMA01WMA11 · Pure Mathematics 1TrigonometryRadians, arcs and sectors

Trigonometry · Subtopic 6 of 8

Radians, Arcs and Sectors

One radian is the angle where the arc equals the radius, so a full turn is $2\pi$ radians. In radians the formulas become beautifully simple: arc length $s=r\theta$ and sector area $A=\tfrac12r^2\theta$ . A segment is a sector minus a triangle.

Edexcel YMA01 WMA11≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Convert between degrees and radians.
Use $s=r\theta$ and $A=\tfrac12r^2\theta$ .
Find the perimeter of a sector.
Find the area of a segment.

Pause and think

Lay the radius of a circle along its edge, as a curved piece of string. How many of these radius-lengths fit around the whole circle?

Section 2 of 14

Core idea

Start hereMeasuring an angle with string

Wrap a piece of string as long as the radius around the edge of a circle. The angle it spans is one radian, about $57.3^\circ$ . Measuring angles this way makes arc length and sector area formulas beautifully simple.

The big idea

$\pi$ radians $=180^\circ$ . Arc $s=r\theta$ ; sector area $A=\frac12r^2\theta$ ( $\theta$ in radians). Segment = sector − triangle.

1. Arc and sector

A sector with angle θ radians

θ (radians) = 0.50

Arc length

=3\theta

, area

=\tfrac12(9)\theta

. At

\theta=2\pi

these become the full circumference

6\pi

and area

9\pi

Fractions of a circle

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

s=r\theta\qquad A=\tfrac12r^2\theta

$r$: radius.
$\theta$: angle in radians.

Worked example $r=3$ , $\theta=1.2$

Arc, area, perimeter of the sector; area of the segment.

Goal: Four values.
Strategy: Formulas; segment = sector − triangle.

$s=3(1.2)=3.6$
Arc.
$A=\tfrac12(9)(1.2)=5.4$
Sector.
$P=3.6+3+3=9.6$
Arc plus two radii.
$\tfrac12(9)\sin1.2=4.19\Rightarrow5.4-4.19=1.21$
Triangle (calculator in radians), subtract.

Answer

3.6

5.4

9.6

1.21

✓ Check

\frac{1.2}{2\pi}=0.191

of the circle:

0.191\times9\pi=5.4

✓.

Common misconceptionCalculator mode

\sin1.2

in degree mode gives

0.0209

, which is wrong here.

Pause and think

Convert $135^\circ$ to radians.

Summary

$\pi=180^\circ$ .
$r\theta$ , $\frac12r^2\theta$ .

Radian: Angle subtending an arc equal to the radius.
Segment: Region between a chord and an arc.

HistoryNaming the radian. Roger Cotes, who worked with Isaac Newton, used the idea of the radian in 1714, but the name "radian" first appeared in print only in 1873, in an exam paper set by James Thomson at Queen's College Belfast.

Fun factWhy calculus prefers radians. Only in radians is the slope of

y=\sin x

at the origin exactly

1

. In degrees it would be

\tfrac{\pi}{180}

, and every calculus formula for trig functions would carry that awkward factor.

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