The central angle

Definition

A central angle is an angle whose vertex is the centre O of the circle, and whose two sides pass through two points on the circle. It ‘points’ from the centre towards two points on the circumference.

This central angle subtends (bounds) a circular arc, that is, the portion of the circle between the two points.

Proportionality between angle and arc

An essential property: the measure of the central angle is proportional to the length of the arc it subtends. The larger the angle, the longer the arc.

A full circle measures 360 degrees and corresponds to the entire circumference.

Example

If a central angle measures 90 degrees (a quarter of a turn), the arc it subtends represents 90 / 360 = 1/4 of the total circumference.

If the circle has a radius r = 8 cm, its total circumference is P = 2 × π × 8 ~ 50.24 cm. The arc subtended by the 90-degree angle therefore measures 50.24 / 4 = 12.56 cm.

Length of an arc: general formula

For a central angle a (in degrees) in a circle of radius r:

length of the arc = (a / 360) × 2 × π × r

Table of notable angles

Central angle	Fraction of the circle	Name of the arc
360 degrees	1 (whole circle)	full circle
180 degrees	1/2	semicircle
90 degrees	1/4	quarter circle

Pitfall to avoid

Do not confuse the central angle (vertex at O) with an angle whose vertex lies on the circle itself: the latter is called an inscribed angle, and this is the subject of the next lesson, which reveals a surprising relationship between the two!