Circle Theorems: Angles at Centre & Circumference

Angle at Centre Theorem

The Angle at Centre Theorem states that the angle formed at the centre of a circle by two points on the circumference is twice the angle formed at the circumference by the same two points.

In other words, if you have an arc on the circle, the angle at the centre subtending that arc is double any angle at the circumference subtending the same arc.

This can be written as:

$$\text{Angle at centre} = 2 \times \text{Angle at circumference}$$

Both angles subtend the same arc of the circle.

This theorem is useful for finding unknown angles in circle problems where you know either the angle at the centre or the angle at the circumference.

For instance, if the angle at the circumference is $30^\circ$, then the angle at the centre is:

$$2 \times 30^\circ = 60^\circ$$

Angle at Circumference Theorem

The Angle at Circumference Theorem states that angles subtending the same arc at the circumference of a circle are equal.

This means if two or more angles are drawn from points on the circumference and they all subtend the same arc, then those angles are equal.

This is sometimes called the angles in the same segment theorem, because these equal angles lie in the same segment of the circle.

This theorem helps to find unknown angles when you know one angle subtending a particular arc.

For example, if two angles subtend the same arc and one is $45^\circ$, then the other must also be $45^\circ$.

Using the Theorems in Angle Calculations

Both the Angle at Centre and Angle at Circumference theorems are often used together to solve problems involving circles.

Remember:

• The angle at the centre is twice any angle at the circumference subtending the same arc.
• Angles subtending the same arc at the circumference are equal.

These facts allow you to find missing angles when given some angles in a circle diagram.

For example, if you know the angle at the centre and one angle at the circumference subtending the same arc, you can check your answers by applying these theorems.

Also, if two angles at the circumference subtend the same arc, they must be equal, which helps to find unknown angles in circle problems.

For example, if you know one angle at the circumference is $40^\circ$, then any other angle subtending the same arc is also $40^\circ$.

These theorems do not apply to angles in semicircles, alternate segments, or cyclic quadrilaterals1those are covered in other topics.

Example: Using the Angle at Centre Theorem

In a circle, the angle at the centre is $100^\circ$. Find the angle at the circumference subtending the same arc.

Since the angle at the centre is twice the angle at the circumference:

$$\text{Angle at circumference} = \frac{100^\circ}{2} = 50^\circ$$