Double the angle
The angle at the centre is double
Two views of an arc
Take an arc AB of a circle. You can look at it from the centre O, or from a point D on the circle, on the far side from the arc. The two views make two different angles.
How does the angle the arc makes at the centre O compare with the angle it makes at D?
What this lesson covers
The idea
The angle subtended by an arc at the centre of a circle is double the angle it subtends at any point on the circle outside the arc.
Two views of an arc
Take an arc AB of a circle. You can look at it from the centre O, or from a point D on the circle, on the far side from the arc. The two views make two different angles.
How does the angle the arc makes at the centre O compare with the angle it makes at D?
- The angle at O is double the angle at D
- The two angles are equal
- The angle at D is double the angle at O
Compare the angles
Drag A, B and D round the circle. The orange arc AXB subtends an angle at the centre O and at D. Try 3 different arcs and read both angles.
Half the angle
The angle subtended by an arc at the centre of a circle is double the angle it subtends at any point on the circle outside the arc.
So ∠AOB = 2 × ∠ADB. Join D to O. Triangles DOA and DOB are isosceles, because OA = OB = OD. In each one, the exterior angle at O is twice the angle at D. Adding the two parts gives ∠AOB = 2∠ADB.
Notes
The angle an arc subtends at the centre is double the angle it subtends at any point on the circle outside the arc.
Check yourself
An arc subtends 80° at the centre of a circle. What angle does it subtend at a point of the circle outside the arc, in degrees?
Answer: 40 °
The angle at the centre is double the angle at the circle, so the angle at the circle is 80° ÷ 2 = 40°.
An arc subtends 35° at a point D of the circle outside the arc. What angle does the arc subtend at the centre, in degrees?
Answer: 70 °
The angle at the centre is double the angle at D: 2 × 35° = 70°.
The major arc AB subtends 260° at the centre O. D is a point on the circle outside this arc, on the minor arc. What is ∠ADB?
An arc subtends ∠ADB = 2x° at a point D of the circle outside the arc, and ∠AOB = 100° at the centre. What is x?
Answer: 25
100° = 2 × (2x°), so 2x = 50 and x = 25.
- 100°. 100° is 360° − 260°, the angle the minor arc subtends at the centre. The angle at D is half of 260°.
- 260°. 260° is the angle at the centre. The angle at D is half of it.
- 130° — correct. Yes. The angle at the circle is half the angle at the centre: 260° ÷ 2 = 130°.