Radian measure, arc length and sector areaEdexcel International A Level Maths: Revision notes

One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. A full turn is $2\pi$ radians, so $\pi\text{ rad}=180^{\circ},\qquad 1\text{ rad}=\frac{180}{\pi}\approx57.3^{\circ}.$ To convert degrees to radians multiply by $\frac{\pi}{180}$; to convert radians to degrees multiply by $\frac{180}{\pi}$. Common values: $30^{\circ}=\frac{\pi}{6}$, $45^{\circ}=\frac{\pi}{4}$, $60^{\circ}=\frac{\pi}{3}$, $90^{\circ}=\frac{\pi}{2}$, $120^{\circ}=\frac{2\pi}{3}$.

For a sector of radius $r$ with angle $\theta$ in radians, the arc length is $s=r\theta.$ Example: $r=6$, $\theta=\frac{2\pi}{3}$ gives $s=4\pi$ cm. Rearranged, $\theta=\frac{s}{r}$: an arc of $10$ cm on a circle of radius $8$ cm subtends $1.25$ radians. The formula is valid only when $\theta$ is in radians.

The area of a sector is $A=\tfrac12r^2\theta,$ with $\theta$ in radians. It is also $A=\frac12rs$, because $s=r\theta$. Example: $r=6$, $\theta=\frac{2\pi}{3}$ gives $A=\frac12(36)\frac{2\pi}{3}=12\pi$. To find an unknown angle, rearrange: if $A=36$ and $r=9$ then $\theta=\frac{2A}{r^2}=\frac89$.

The boundary of a sector is two radii and the arc: $P=2r+r\theta=r(2+\theta).$ Example: $r=6$, $\theta=\frac{2\pi}{3}$ gives $P=12+4\pi=24.6$ cm. When the perimeter and one other quantity are given, form an equation in $r$ and $\theta$ and solve.

The minor segment between a chord and its arc has area equal to the sector minus the triangle: $\tfrac12r^2\theta-\tfrac12r^2\sin\theta=\tfrac12r^2(\theta-\sin\theta).$ For $r=10$, $\theta=1.2$: $60-46.6=13.4$. The chord is $2r\sin\frac{\theta}{2}$, so the perimeter of the segment is $r\theta+2r\sin\frac{\theta}{2}$. Keep the calculator in radian mode and give exact answers (in terms of $\pi$) when the question asks.