A-level Mathematics/CIE/Pure Mathematics 1/Circular Measure

Defining the radian


A radian is a unit for measuring angles. It is defined as the angle subtended by an arc that is as long as the radius. As a consequence of this, there are $$2\pi$$ radians in a full circle, because the length of the circumference is $$2\pi$$ times the length of the radius.

Converting between radians and degrees
Degrees are another common unit for measuring angles. There are $$360^{o}$$ in a circle, thus $$360^{o} = 2\pi\text{rad}$$.

To convert from degrees to radians, multiply the number of degrees by $$\frac{\pi}{180}$$.

e.g. $$30^{o}$$ is equal to $$30\times\frac{\pi}{180} = \frac{\pi}{6}$$

To convert from radians to degrees, multiply the amount of radians by $$\frac{180}{\pi}$$.

e.g. $$\frac{\pi}{3}$$ radians is equal to $$\frac{\pi}{3}\times\frac{180}{\pi} = \frac{180}{3} = 60^{o}$$

Arc length


Arc length is, unsurprisingly, the length of a circular arc. This length depends on the size of the radius and the angle that the arc subtends.

We can think of an arc as a fraction of the circumference $$2\pi r$$. This means that the arc length is the angle divided by a full circle times the length of the circumference: $$s = \frac{\theta}{2\pi}(2\pi r)$$ which can be simplified to $$s = r\theta$$.

e.g. The arc length for an arc with radius $$2$$ and angle $$2\text{rad}$$ is $$2(2) = 4$$.

Sector areas
The area of a sector can be derived in a similar way: it is a fraction of the area of a circle. The area of a circle is $$\pi r^2$$, so the area of a sector is $$\frac{\theta}{2\pi}\pi r^2 = \frac{r^2\theta}{2}$$.