Radians, arcs and sectors

A radian measures an angle using the circle itself. One radian is the angle subtended at the centre when the arc length equals the radius.

Because a complete circumference has length $2\pi r$, a full turn contains $2\pi$ radians:

$360^\circ=2\pi\text{ radians}.$

Therefore,

$180^\circ=\pi\text{ radians}.$

Converting angles

To convert degrees to radians, multiply by $\pi/180$:

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

To convert radians to degrees, multiply by $180/\pi$.

Arc length

When $\theta$ is measured in radians, the length of an arc is

$s=r\theta.$

For radius $6$ cm and angle $\pi/4$,

$s=6\times\frac{\pi}{4}=\frac{3\pi}{2}\text{ cm}.$

This compact formula works precisely because radians compare the arc length directly with the radius.

Area of a sector

The corresponding sector area is

$A=\frac12r^2\theta.$

With $r=6$ and $\theta=\pi/4$,

$A=\frac12(6^2)\frac{\pi}{4}=\frac{9\pi}{2}\text{ cm}^2.$

Common mistakes

• The formulae $s=r\theta$ and $A=\tfrac12r^2\theta$ require radians.
• Keep exact multiples of $\pi$ until a decimal is requested.
• Check whether a perimeter includes two radii as well as the arc.
• Calculator trigonometry questions may require radian mode; verify the display before starting.