Angles standing on the same arc
An arc of a circle can be 'seen' from the centre and from the edge. The central angle has its vertex at the centre O; the angle at the circumference has its vertex on the circle. Both stand on the same arc.
Key idea
Central angle = 2 × angle at the circumference (same arc). So angle at the circumference = ½ × central angle.
Why it helps
If the central angle on arc AB is 80°, then any angle at the circumference standing on AB is 80 ÷ 2 = 40°. Turned around, a circumference angle of 30° means the central angle is 2 × 30 = 60°.
Worked example
Arc PQ gives a central angle POQ = 130°. The angle PRQ at the circumference (R on the major arc) = 130 ÷ 2 = 65°. A second point S on the same arc also gives angle PSQ = 65°, because angles on the same arc are equal.
Angle in a semicircle
When the arc is a semicircle, the central angle is a straight line = 180°, so the angle at the circumference = 180 ÷ 2 = 90°. This is the 'angle in a semicircle' fact, and it means any triangle drawn with the diameter as one side and its third point on the circle is always right-angled.
When you meet a circle problem, first find the arc that both angles rest on, then decide which angle sits at the centre. Doubling or halving between them solves most of these questions in one step.
Remember
- Both angles must stand on the SAME arc.
- Angles at the circumference on the same arc are equal.
- Angle in a semicircle = 90°.