Angles at Centre & Circumference

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Angles at centre & circumference

Circle Theorems deal with angles that occur when lines are drawn within (and connected to) a circle

You may need to use other facts and rules such as:
- basic properties of lines and angles
- properties of triangles and quadrilaterals
- angles in parallel lines or polygons

In this theorem, the chords (radii) to the centre and the chords to the circumference are both drawn from (subtended by) the ends of the same arc

To spot this circle theorem on a diagram
- find any two radii in the circle and follow them to the circumference
- see if there are lines from those points going to any other point on the circumference
- it may look like the shape of an arrowhead
When explaining this theorem in an exam you must use the keywords:
- The angle at the centre is twice the angle at the circumference

This theorem is still true when the ‘triangle parts’ overlap
It is also true when the lines form a diamond shape
- You need to compare the reflex angle at the centre with the angle at the circumference
- Common mistakes are to
  - compare the wrong angles
  - confuse it with a different circle theorem on cyclic quadrilaterals

Worked example

Find the value of $x$ in the diagram below.

Not drawn accurately

Give a reason for each step of your working.

Answer:

There are three radii in the diagram, AO, BO and CO
Mark these as equal length lines

Notice how they create two isosceles triangles
Base angles in isosceles triangles are equal

Angle OAB = angle OBA = 60º (isosceles triangle)

Use the circle theorem:

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

Form an equation for $x$

$2 (x + 60) = 150$

Expand the brackets and solve the equation

$\begin{array}{rcl} 2 x + 120 & = & 150 \\ 2 x & = & 30 \\ x & = & 15 \end{array}$

$x = 15$

Base angles in isosceles triangles are equal

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