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