3.4 Radians, arcs and sectorsIB Maths: Analysis and Approaches SL: 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 $\pi\text{ radians}=180^\circ.$

• Degrees to radians: multiply by $\frac{\pi}{180}$. So $150^\circ=\frac{5\pi}{6}$ and $120^\circ=\frac{2\pi}{3}$.
• Radians to degrees: multiply by $\frac{180}{\pi}$. So $0.3$ rad $=17.2^\circ$ and $1$ rad $\approx57.3^\circ$. Radian measure can be written as an exact multiple of $\pi$ or as a decimal such as $0.3$ or $1.2$. An angle with no degree sign is in radians.

For a sector with radius $r$ and angle $\theta$ in radians: $l=r\theta,\qquad A=\frac12r^2\theta.$ These come from taking the fraction $\frac{\theta}{2\pi}$ of the circumference $2\pi r$ and of the area $\pi r^2$. The perimeter of a sector is $2r+r\theta$ — both radii plus the arc. Example: $r=6$, $\theta=\frac{5\pi}{6}$: arc $5\pi$, area $15\pi$, perimeter $12+5\pi$.

A segment is the region between a chord and an arc. For the minor segment cut off by a chord that subtends $\theta$ at the centre: $\text{segment area}=\frac12r^2\theta-\frac12r^2\sin\theta=\frac12r^2(\theta-\sin\theta).$ The chord length comes from the cosine rule in the isosceles triangle $OAB$: $AB^2=2r^2-2r^2\cos\theta$. The perimeter of the segment is chord plus arc. Example: $r=10$, $\theta=\frac{2\pi}{3}$ gives area $\frac{100\pi}{3}-25\sqrt3$ and perimeter $\frac{20\pi}{3}+10\sqrt3$.

Harder questions give the area and perimeter (or arc) and ask for $r$ and $\theta$. Write one equation for each fact, then eliminate $\theta$:

• Area 50: $\frac12r^2\theta=50\Rightarrow r^2\theta=100$.
• Perimeter 30: $2r+r\theta=30\Rightarrow r\theta=30-2r$.
• Substitute: $r(30-2r)=100\Rightarrow r^2-15r+50=0$, so $r=5$ or $10$. Then check each solution in context: here $\theta=4$ or $\theta=1$, and both are less than $2\pi$, so both are valid sectors.