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

Radians

Defining the radian

The radian is defined as an angle subtended by an arc the same length as the radius.

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 $\frac{\pi }{3}\times \frac{180}{\pi }=\frac{180}{3}=60^o$

Geometric Calculations

Arc length

The length L is the arc. The area in green is the sector.

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 $\frac{\theta }{2\pi }\pi r^2=\frac{r^2\theta }{2}$.