An arc makes twice the angle at the centre that it makes at the edge
One number matters again and again in this chapter. It is 2.
- Points A and B on a circle, centre O.
- A third point P, on the rest of the circle.
- Angle AOB, at the centre, standing on arc AB.
- Angle APB, at P, standing on the same arc.
- The centre angle is double the rim angle.
The rule
Join A and B to the centre O. That gives the angle at the centre. Join A and B to a point P instead. P sits on the rest of the circle, away from arc AB. That gives the angle at the edge, also called the rim angle. Both angles stand on arc AB. The centre angle is double the edge angle on the same arc.
Why this matters
Take P so that the centre O is inside angle APB. OA and OP are radii. So triangle OAP has equal angles at A and P. Call each x. Extend PO past O. The new angle at O is outside triangle OAP. It equals the angles at A and P added. So it is 2x. In triangle OBP, call the equal angles y. Its new angle at O is 2y. So angle AOB = 2x + 2y = 2(x + y). Angle APB = x + y. So angle AOB is double angle APB. Other positions of P need a similar argument.
| Angle at the centre | Angle at the rim | Check |
|---|---|---|
| 100° | 50° | 2 × 50 = 100 |
| 130° | 65° | 2 × 65 = 130 |
| 180° | 90° | 2 × 90 = 180 |
| 260° | 130° | 2 × 130 = 260 |
The last row's arc is more than half the circle. Its 260° is measured the long way round O. The rule still holds there.
Open the arc. Compare the two angles.
Loading lab…