Central Angle Theorem

Theorem: The central angle subtended by two points on a circle is twice the inscribed angle subtended by those points.
Try this Drag the orange dot at point P. Note that the central angle AOB is always twice the inscribed angle APB.

The Central Angle Theorem states that the measure of inscribed angle (APB) is always half the measure of the central angle AOB. As you adjust the points above, convince yourself that this is true.


This theorem only holds when P is in the major arc. If P is in the minor arc (that is, between A and B) the two angles have a different relationship. In this case, the inscribed angle is the supplement of half the central angle. As a formula: In other words, it is 180 minus what it would normally be.

Other circle topics


Equations of a circle

Angles in a circle