Angles at the centre and circumference - Higher
The angle An angle created by an object at a given point. by an The curve between two points on the circumference of a circle. at the centre is twice the angle subtended at the Circumference is the name of the perimeter of a complete circle, that is, the distance all around it..
More simply, the angle at the centre is double the angle at the circumference.
Example
Calculate the missing angles \(x\) and \(y\).
\(x\) = \(50 \times 2 = 100^\circ\)
\(y\) = \(40 \times 2 = 80^\circ\)
Proof
Let angle OGH = \(y\) and angle OGK = \(x\).
Angle OGH (\(y\)) = angle OHG because triangle GOH is an Two sides have equal lengths. Angles opposite the equal sides are equal.. Lengths OH and OG are both The distance from the centre of a circle to its circumference. The plural of radius is radii..
Angle OGK (\(x\)) = angle OKG because triangle GOK is also isosceles. Lengths OK and OG are also both radii.
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°)
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°)
The angle at the centre KOH (\(2y + 2x\)) is double the angle at the circumference KGH (\(x + y\)).