Sectors and Arcs

A slice of a circle

A sector is a "pizza slice" of a circle — the region bounded by two radii and the curved arc between them. The angle at the centre, between the two radii, is \theta ("theta").

A full turn is 360^\circ, so a sector is just the fraction \frac{\theta}{360} of the whole circle. Take that same fraction of the circumference to get the arc length:

\frac{\theta}{360} \times 2\pi r

and that same fraction of the area to get the sector area:

\frac{\theta}{360} \times \pi r^2

Half and quarter circles

The fraction \frac{\theta}{360} makes the common slices easy: a semicircle is \theta = 180^\circ (so \frac{180}{360} = \tfrac12, half), and a quarter circle is \theta = 90^\circ (so \frac{90}{360} = \tfrac14).

a sector is \frac{\theta}{360} of the whole circle;
arc length = \frac{\theta}{360} \times the circumference (2\pi r);
sector area = \frac{\theta}{360} \times \pi r^2;
a semicircle is \theta = 180^\circ (half); a quarter circle is \theta = 90^\circ.

See a sector on a circle

Step through the figure: two radii open out an angle \theta, then the slice between them is shaded — here a quarter of the circle.