Theorem

The angle at the centre is twice the angle at the circumference

Geometry • Circle • Arcs and Angles

Formula

∠ B O C = 2∠ B A C

An arc

B C

of a circle with centre

O

subtends at

O

an angle twice that which it subtends at any point

A

on the remaining part of the circle:

∠ B O C = 2∠ B A C

. Two corollaries follow: angles in the same segment are equal, and an angle in a semicircle is

9 0^{\circ}

How to use it

1Join the centre to the two ends of the arc: the triangle formed is isosceles, with the two radii as its equal sides.
2Use it in reverse to get the angle at the circumference when the central angle is known, halving instead of doubling.
3When the chord is a diameter the central angle is $18 0^{\circ}$ , so the angle at the circumference is $9 0^{\circ}$ .

A

B

and

C

lie on a circle with centre

O

, and

∠ B A C = 3 5^{\circ}

. What is

∠ B O C

Step-by-step

1
$∠ B O C = 2∠ B A C$
Why: The arc $B C$ subtends $∠ B O C$ at the centre and $∠ B A C$ at the point $A$ of the remaining arc.
2
$∠ B O C = 2 \times 3 5^{\circ} = 7 0^{\circ}$

Answer

7 0^{\circ}

Watch out

The point

A

must lie on the arc LEFT OVER by

B

and

C

; if

A

is on the same arc as the chord's minor side, the doubled angle is the reflex angle at the centre, and the ordinary angle

∠ B O C

is then

36 0^{\circ}

minus it.

Builds on

Also known as

angle at the centre, inscribed angle theorem, angle subtended by an arc, angles in the same segment, angle in a semicircle