Angles at the centre and circumference - Higher

The angle subtended by an arc at the centre is twice the angle subtended at the circumference.

Angle at centre is double the corresponding angle at the circumference

More simply, the angle at the centre is double the angle at the circumference.

Example

Calculate the missing angles x and y.

Unknown and known angle at centre and circumference

x = 50 \times 2 = 100^\circ

y = 40 \times 2 = 80^\circ

Proof

Let angle OGH = y and angle OGK = x.

Circle with a triangle either side of centre line, with unknown angles, x and y at the same point on the circumference.

Angle OGH ( y) = angle OHG because triangle GOH is an isosceles. Lengths OH and OG are both radii.

Angle OGK ( x) = angle OKG because triangle GOK is also isosceles. Lengths OK and OG are also both radii.

Circle with a triangle either side of centre line, with unknown angles, x and y at the same point on the circumference and also at opposite sides of the circumference.

Angle GOH = 180 - 2y (because angles in a triangle add up to 180°)

Angle GOK = 180 - 2x (because angles in a triangle add up to 180°)

Circle with triangle either side of centre line, unknown angles x and y at same point on circumference, also at opposite sides of circumference. Angles at centre labelled (inside triangles)

Angle JOH = 2y (because angles on a straight line add up to 180°. 180 - 2y + 2y = 180)

Angle JOK = 2x (because angles on a straight line add up to 180°)

Circle with triangle either side of centre line, unknown angles x and y at same point on circumference, and at opposite sides of circumference. Angles at the centre (outside triangles) labelled 2y.

The angle at the centre KOH ( 2y + 2x) is double the angle at the circumference KGH ( x + y).