Angles at the centre and circumference

The angle at the centre is twice the angle at the circumference standing on the same arc. The circumference angle’s vertex lies on the circle, and its arms join the two endpoints of that arc. Use the arc not containing the circumference vertex.

Example

If the minor central angle \(AOB\) is \(120^\circ\), an angle \(APB\) standing on that minor arc is \(60^\circ\). For a vertex on the minor arc, the angle instead stands on the major arc; use reflex central angle \(240^\circ\), giving a circumference angle of \(120^\circ\).

Identify both arc endpoints before doubling or halving. Two angles that merely share the letters \(A\) and \(B\) may stand on different arcs.