Arc length and sector area in radians

For \(\theta\) in radians, the sector is \(\dfrac\theta{2\pi}\) of a circle. Hence \(l=r\theta\) and \(A=\dfrac12r^2\theta=\dfrac12rl\). These compact formulas require radians.

Example

A sector has radius 8 cm and angle 1.5 radians. Its arc length is 12 cm, area is 48 cm² and perimeter is \(16+12=28\text{ cm}\). For the corresponding major sector, use angle \(2\pi-1.5\), not 1.5.

The formulas can work backwards: if arc length is 10 cm and sector area is 35 cm², \(35=\dfrac12r(10)\), so \(r=7\text{ cm}\) and \(\theta=\dfrac{10}{7}\) radians. Check that the sector uses the same radius and arc in both equations.

Keep \(\pi\) exact when required and keep full calculator precision between steps. Trace the requested region: a major sector is a valid sector and its angle exceeds \(\pi\) radians.