What is the relationship between the angle at the centre and the angle at the circumference?

The angle at the centre is **twice** the angle at the circumference when both are subtended by the same arc.

What shape does this theorem often form on a diagram?

An **arrowhead** shape — two radii from the centre to the circumference, with lines from those points meeting at another point on the circumference.

When the lines form a diamond shape, which angle at the centre do you use?

The **reflex angle** at the centre — the larger angle (greater than 180°) — must be compared with the angle at the circumference.

Why do radii often create isosceles triangles?

Because all radii of a circle are **equal in length**, so any triangle with two radii as sides has two equal sides, making it isosceles.

What is a common mistake when using this theorem?

**Comparing the wrong angles** — ensure you identify which angle is at the centre (between two radii) and which is at the circumference (between chords from the arc endpoints).

What does 'subtended by the same arc' mean?

It means the angles are formed by lines drawn from the **same two points** on the circumference — the chord (or arc) between those points 'subtends' (creates) both angles.

If the angle at the centre is 120°, what is the angle at the circumference?

**60°** — because the angle at the circumference is half the angle at the centre (120° ÷ 2 = 60°).

How does this theorem relate to the angle in a semicircle?

The angle in a semicircle (90°) is a **special case** — the angle at the centre on a diameter is 180°, so the angle at the circumference is 180° ÷ 2 = 90°.

What exam keywords must you use when explaining this theorem?

**The angle at the centre is twice the angle at the circumference** — use this exact phrase for full marks.

Angles at Centre & Circumference | Pearson Edexcel IGCSE Maths

Q: If the angle at the circumference is 35°, what is the angle at the centre?

**70°** — because the angle at the centre is twice the angle at the circumference (2 × 35° = 70°).

Geometry & Trigonometry

Angles at Centre & Circumference

Summary

The angle at the centre is twice the angle at the circumference. This fundamental circle theorem states that when chords (radii) to the centre and chords to the circumference are both drawn from the ends of the same arc, the angle subtended at the centre is exactly double the angle subtended at the circumference. It applies to various configurations including arrowhead and diamond shapes, and is essential for solving angle problems in circle geometry.

1. The Theorem Statement

Circle Theorem: The angle at the centre is twice the angle at the circumference.

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.

Circle theorems deal with angles that occur when lines are drawn within and connected to a circle. You may need to use other facts: basic properties of lines and angles, properties of triangles and quadrilaterals, and angles in parallel lines or polygons.

2. Diagram: Basic Configuration

The diagram shows centre O with points A, B, C on the circumference. Angle at centre (AOB) = 2 × angle at circumference (ACB).

3. How to Spot This Theorem

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 in an exam, use the keywords: The angle at the centre is twice the angle at the circumference.

4. Special Cases

This theorem is still true when the 'triangle parts' overlap.

It is also true when the lines form a diamond shape.

For diamond shapes, compare the reflex angle at the centre with the angle at the circumference.

Common mistakes: comparing the wrong angles, or confusing it with the cyclic quadrilateral theorem.

5. Worked Example

6. Examiner Tips

Angles at Centre & Circumference

Summary

1. The Theorem Statement

Circle Theorem: The angle at the centre is twice the angle at the circumference.

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.

2. Diagram: Basic Configuration

The diagram shows centre O with points A, B, C on the circumference. Angle at centre (AOB) = 2 × angle at circumference (ACB).

3. How to Spot This Theorem

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 in an exam, use the keywords: The angle at the centre is twice the angle at the circumference.

4. Special Cases

This theorem is still true when the 'triangle parts' overlap.

It is also true when the lines form a diamond shape.

For diamond shapes, compare the reflex angle at the centre with the angle at the circumference.

Common mistakes: comparing the wrong angles, or confusing it with the cyclic quadrilateral theorem.

5. Worked Example

Problem: Find the value of x. There are three radii AO, BO, CO creating two isosceles triangles. Base angles in isosceles triangles are equal, so angle OAB = angle OBA = 60°.

Using the theorem: 2(x + 60) = 150 → 2x + 120 = 150 → 2x = 30 → x = 15.

Reasons: Base angles in isosceles triangles are equal; The angle at the centre is twice the angle at the circumference.

6. Examiner Tips

Questions often ask for 'reasons' — quote an angle fact or circle theorem for every angle you find, not just the final answer.

Mark equal radii on the diagram to identify isosceles triangles.

Form equations systematically: centre angle = 2 × circumference angle.