Sweep round the centre

The idea

The angle subtended by arc AB at the centre O is the angle swept moving from OA to OB along the arc; the minor arc subtends ∠AOB and the major arc the angle swept the other way.

Swing your arm

Stand at the centre O of a circle and point one arm at A. Now swing your arm round to point at B. The angle you swept depends on which way you swing.

You swing from A to B the short way and then the long way. What do the two swept angles add up to?

• A full turn, 360°
• A straight angle, 180°
• They are not related

Sweep from A to B

Drag the point P round the circle, starting at A. The arm OP sweeps an angle. Stop on B. Do it the short way, then the long way.

Two arcs, two angles

The angle subtended by an arc AB at the centre O is the angle swept as you move from OA to OB along the arc. The minor arc subtends ∠AOB; the major arc subtends the angle swept the other way round.

Here the minor arc subtends 120° and the major arc subtends 240°. Together they make a full turn: 120° + 240° = 360°.

Notes

The angle an arc subtends at the centre is the angle swept from OA to OB along the arc: the minor arc subtends ∠AOB, and the major arc the angle swept the other way.

Check yourself

The minor arc AB of a circle subtends 100° at the centre O. What angle does the major arc AB subtend at O, in degrees?

Answer: 260 °

The two angles add up to a full turn: 360° − 100° = 260°.

The major arc AB subtends 250° at the centre. What angle does the minor arc AB subtend?

A and B are the two ends of a diameter of a circle with centre O. What angle does each of the two arcs AB subtend at O, in degrees?

Answer: 180 °

AB passes through O, so ∠AOB is a straight angle. Each arc is half the circle and subtends 180°.

What is the angle subtended by an arc AB at the centre O?

• 110° — correct. Yes. 360° − 250° = 110°.
• 250°. That is the angle the major arc subtends. The minor arc subtends the rest of the full turn.
• 180°. Both arcs subtend 180° only when AB is a diameter. Here the major arc subtends 250°, so the minor arc subtends 360° − 250°.
• The angle between the chord AB and the radius OA. That is an angle at A, not at the centre.
• The angle swept as you move from OA to OB along the arc — correct. Yes. You follow the arc from A to B and measure the angle that OA turns through.
• The angle at B in the triangle OAB. That is an angle at B, not at the centre O.