Area and Arc Length of a Sector

A sector is a slice of a circle. Its area is π × radius² × angle ÷ 360, and its arc length is 2 × π × radius × angle ÷ 360.

Explanation

A sector is the slice of a circle cut off by two radii. The central angle says what fraction of the full $360^\circ$ the slice covers, and both the area and the arc length are that same fraction of the whole circle.

S = \pi r^2 \times \dfrac{\theta}{360} \qquad \ell = 2 \pi r \times \dfrac{\theta}{360}

$S$ — the area of the sector
$\ell$ — the arc length, the curved part
$r$ — the radius
$\theta$ — the central angle, in degrees

The perimeter is the arc plus the two straight radii: $L = \ell + 2r$ .

Example

With the defaults, a radius of $r = 6$ and a central angle of $\theta = 60^\circ$ . Since $\dfrac{60}{360} = \dfrac{1}{6}$ , the sector is one sixth of the circle.

area $S = \pi \times 6^2 \times \dfrac{1}{6} = 6\pi \approx 18.8496$
arc length $\ell = 2\pi \times 6 \times \dfrac{1}{6} = 2\pi \approx 6.2832$
perimeter $L = 2\pi + 2 \times 6 \approx 18.2832$

Watch out

The perimeter is not the arc on its own. Remember to add the two radii, which is what this calculator does: $\ell + 2r$ . Enter the angle in degrees; anything outside the range $0^\circ$ to $360^\circ$ is rejected, and the radius must be positive. At $360^\circ$ the sector becomes the whole circle, and the formulas collapse to $\pi r^2$ and $2\pi r$ .