Mathematics 9709 · Pure Mathematics 1

Length of an arc

An arc is a part of the circle's edge. With the angle in radians (§4.1), its length has the simplest formula you meet all year.
The mark people often lose is on the perimeter of a sector, so we cover that too.

Arc length is $s=r\theta$ with $\theta$ in radians. A sector's perimeter is the arc plus the two straight radii: $P=r\theta+2r$ , not the arc on its own.

Arc length is just rθ

An angle of 1 radian makes an arc of length $r$ . So $\theta$ radians makes $\theta$ radius-lengths, which is $r\theta$ .
The arc grows in step with the angle: no $\pi$ , no division.

$s=r\theta\qquad(\theta\text{ in radians})$

The arc AB has length rθ (gold). The two straight sides are each the radius r.

Why it scales: 1 rad → r, so θ rad → rθ

The whole edge is $2\pi r$ , and a full turn is $2\pi$ radians. So one radian of turn gives $\dfrac{2\pi r}{2\pi}=r$ of arc. That is one radius-length. Two radians give $2r$ , three give $3r$ , and $\theta$ radians give $r\theta$ . This neat link is the whole reason radians exist. In degrees the same arc would be $\dfrac{\theta}{360}\times2\pi r$ , and that is why we change to radians first.

Worked examples

Worked example

Arc subtending $\tfrac{\pi}{3}$ rad in a circle of radius $15$ cm

The angle is already in radians, so put it straight in:

$s=r\theta=15\times\dfrac{\pi}{3}=5\pi\text{ cm}$

Answer

s=5\pi\text{ cm}\approx15.7\text{ cm}

Worked example

Perimeter of a sector: radius $10$ cm, angle $72^\circ$

The angle is in degrees, so change it to radians first (this is the trap), then the arc and the perimeter are two short steps:

Convert the angle to radians (every formula here needs it):
$\theta=72\times\dfrac{\pi}{180}=\dfrac{2\pi}{5}\text{ rad}$
Arc length $s=r\theta$ :
$s=10\times\dfrac{2\pi}{5}=4\pi\text{ cm}$
Perimeter = arc + the two straight radii, the step people forget:
$P=r\theta+2r=4\pi+2(10)=4\pi+20$

Answer

P=4\pi+20\approx32.6\text{ cm}

Where the marks go

Common mistake

s=r\theta

needs

\theta

in radians. Putting in

72

instead of

\tfrac{2\pi}{5}

gives a very wrong arc. Change degrees to radians first, every time.

Common mistake

The perimeter of a sector is

P=r\theta+2r

: the arc and the two radii. Writing just the arc loses an easy mark.

Now you try

A sector $OAB$ of a circle of radius $r$ cm has arc length $9.6$ cm and area $76.8\text{ cm}^2$ . Find $r$ and the angle $\theta$ . (9709/13 May/Jun 2023 Q6)

Your turn— tap to reveal the worked answer (9709/13 May/Jun 2023 Q6)

Use both formulas: arc $=r\theta$ and area $=\tfrac12 r^2\theta$ (the sector area, §4.3). Notice that $\tfrac12 r^2\theta=\tfrac12 r(r\theta)$ , and $r\theta$ is the arc you already know:

$76.8=\tfrac12\,r\,(r\theta)=\tfrac12\,r\,(9.6)$

So $r=\dfrac{76.8\times2}{9.6}=16$ , then divide the arc by it:

$\theta=\dfrac{9.6}{16}=0.6$

Answer

r=16\text{ cm},\ \theta=0.6\text{ rad}

PreviousRadians and converting NextArea of a sector and a segment

Length of an arc

An arc is a part of the circle's edge. With the angle in radians (§4.1), its length has the simplest formula you meet all year.
The mark people often lose is on the perimeter of a sector, so we cover that too.

Arc length is $s=r\theta$ with $\theta$ in radians. A sector's perimeter is the arc plus the two straight radii: $P=r\theta+2r$ , not the arc on its own.

Arc length is just rθ

An angle of 1 radian makes an arc of length $r$ . So $\theta$ radians makes $\theta$ radius-lengths, which is $r\theta$ .
The arc grows in step with the angle: no $\pi$ , no division.

$s=r\theta\qquad(\theta\text{ in radians})$

The arc AB has length rθ (gold). The two straight sides are each the radius r.

Why it scales: 1 rad → r, so θ rad → rθ

Worked examples

Worked example

Arc subtending $\tfrac{\pi}{3}$ rad in a circle of radius $15$ cm

The angle is already in radians, so put it straight in:

$s=r\theta=15\times\dfrac{\pi}{3}=5\pi\text{ cm}$

Answer

s=5\pi\text{ cm}\approx15.7\text{ cm}

Worked example

Perimeter of a sector: radius $10$ cm, angle $72^\circ$

The angle is in degrees, so change it to radians first (this is the trap), then the arc and the perimeter are two short steps:

Convert the angle to radians (every formula here needs it):
$\theta=72\times\dfrac{\pi}{180}=\dfrac{2\pi}{5}\text{ rad}$
Arc length $s=r\theta$ :
$s=10\times\dfrac{2\pi}{5}=4\pi\text{ cm}$
Perimeter = arc + the two straight radii, the step people forget:
$P=r\theta+2r=4\pi+2(10)=4\pi+20$

Answer

P=4\pi+20\approx32.6\text{ cm}

Where the marks go

Common mistake

s=r\theta

needs

\theta

in radians. Putting in

72

instead of

\tfrac{2\pi}{5}

gives a very wrong arc. Change degrees to radians first, every time.

Common mistake

The perimeter of a sector is

P=r\theta+2r

: the arc and the two radii. Writing just the arc loses an easy mark.

Now you try

A sector $OAB$ of a circle of radius $r$ cm has arc length $9.6$ cm and area $76.8\text{ cm}^2$ . Find $r$ and the angle $\theta$ . (9709/13 May/Jun 2023 Q6)

Your turn— tap to reveal the worked answer (9709/13 May/Jun 2023 Q6)

Use both formulas: arc $=r\theta$ and area $=\tfrac12 r^2\theta$ (the sector area, §4.3). Notice that $\tfrac12 r^2\theta=\tfrac12 r(r\theta)$ , and $r\theta$ is the arc you already know:

$76.8=\tfrac12\,r\,(r\theta)=\tfrac12\,r\,(9.6)$

So $r=\dfrac{76.8\times2}{9.6}=16$ , then divide the arc by it:

$\theta=\dfrac{9.6}{16}=0.6$

Answer

r=16\text{ cm},\ \theta=0.6\text{ rad}