The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference. (Abbreviation: angle at centre = 2 angles at circumference)

Example 1:

In a diagram, O is the centre of the circle with points A, P, B touching the sides of the circle. Angle APB = 47 degrees. Find angle AOB.













Angle AOB = 2 angle APB (Abbreviation: angle at centre = 2 angles at circumference)
= 2 x 47 degrees
= 94 degrees

Example 2:

In a diagram, O is the centre of the circle with points A, P, B touching the sides of the circle. Angle AOB = 110 degrees. Find angle APB.















Angle AOB = 2 angle APB (Abbreviation: angle at centre = 2 angles at circumference)
= 110 degrees divided 2
= 55 degrees

Example 3:

In a diagram, O is the centre of the circle with points A, P, B touching the sides of the circle. Angle AOB = 118 degrees. Find angle APB.

Reflex angle AOB = 360 degrees – 118 degrees (angles at a point)
= 242 degrees
Reflex angle AOB = 2 angles APB
242 degrees = 2 angles APB
Angle APB = 121 degrees