Centre and circumference angles

Objective. Use the doubled central-angle relationship.

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Learn: the key idea

The centre and circumference angles theorem states that the angle subtended at the centre of a circle by an arc is exactly twice the angle subtended at the circumference by the same arc. Identify the arc shared by both angles, then either double the circumference angle to find the centre angle, or halve the centre angle to find the circumference angle. If the angle at the centre is 100°, the corresponding angle at the circumference on the same arc is 50°.

Worked example

A chord subtends an angle of 38° at the circumference. Use the centre and circumference angle relationship to find the angle subtended at the centre by the same arc.

1 $Identify that both angles stand on the same arc$
2 $Recall the rule: centre angle = 2 \times circumference angle$
3 $Substitute: centre angle = 2 \times 38°$
4 $Calculate: centre angle = 76°$

Answer: 76°

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Common mistake. Students sometimes apply the doubling rule to angles that do not share the same arc, giving an incorrect relationship. Always confirm both angles stand on the same arc before doubling or halving.

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