Circle theorems are a crucial part of the GCSE Maths curriculum across all exam boards (AQA, Edexcel, OCR). They often appear in the higher-tier papers and can be the key to unlocking several marks. This guide breaks down all eight theorems in a clear, logical way, perfect for UK teachers to use in lessons and for Year 10–11 students revising for their exams.
What Are Circle Theorems?
Circle theorems are rules about the angles and lines you can create inside and outside a circle. They allow you to calculate missing angles without needing a protractor, using pure logic and geometry. Mastering them involves two key skills: identifying which theorem to use and applying it correctly.
The Eight Circle Theorems: A Step-by-Step Breakdown
Theorem 1: The Angle at the Centre
Rule: The angle subtended at the centre is twice the angle subtended at the circumference by the same arc.
How to spot it: Look for two angles—one at the centre (O) and one at the circumference (A, B, or C)—that are both created from the same two points on the circle (the same arc, like arc BC).
> Formula: Angle at centre = 2 × Angle at circumference.
Theorem 2: Angles in the Same Segment
Rule: Angles subtended by the same arc in the same segment are equal.
How to spot it: Identify points on the circumference that form a chord. Any triangle drawn using these two points and a third point on the same arc of the circle will have the same angle at that third point.
Theorem 3: The Angle in a Semicircle
Rule: The angle in a semicircle is a right angle (90°).
How to spot it: The three points of the triangle must form a diameter (a straight line through the centre). The angle opposite this diameter is always 90°.
Theorem 4: Opposite Angles in a Cyclic Quadrilateral
Rule: Opposite angles in a cyclic quadrilateral (a four-sided shape where all vertices touch the circle) add up to 180°.
How to spot it: A quadrilateral drawn inside a circle, with all four corners on the circumference.
> Formula: Angle A + Angle C = 180°, Angle B + Angle D = 180°.
Theorem 5: The Tangent and Radius Theorem
Rule: A tangent to a circle meets the radius at the point of contact at 90°.
How to spot it: A straight line (tangent) that just touches the circle, and a radius drawn to that exact point.
Theorem 6: Tangents from the Same Point
Rule: The two tangents drawn from a single external point to a circle are equal in length.
How to spot it: Two lines from a point outside the circle that both just touch the circle.
Theorem 7: The Alternate Segment Theorem
Rule: The angle between a tangent and a chord is equal to the angle in the alternate segment.
How to spot it: First, find a tangent touching the circle. Then, draw a chord from that point. The angle between the tangent and the chord equals the angle on the opposite side of the chord, inside the circle.
Theorem 8: Perpendicular Line from Centre to Chord
Rule: A perpendicular line from the centre of a circle to a chord bisects the chord (cuts it exactly in half).
How to spot it: A line from the centre meeting a chord at a 90° angle.
Your Problem-Solving Strategy: A Flowchart
The hardest part is knowing where to start. Use this simple decision flowchart to guide your thinking.
flowchart TD
A["Start: Identify all angles and lines"] --> B["Is there a tangent?"]
B -->|Yes| C["Which tangent theorem?"]
C --> D["Tangent and Radius: Look for 90° angle"]
C --> E["Two Tangents: Look for equal lengths"]
C --> F["Alternate Segment: Angle between
tangent and chord"]
B -->|No| G["Is there a cyclic quadrilateral?"]
G -->|Yes| H["Opposite angles add to 180°"]
G -->|No| I["Is there a diameter/semicircle?"]
I -->|Yes| J["Angle in a semicircle = 90°"]
I -->|No| K["Is there an angle at the centre?"]
K -->|Yes| L["Angle at centre = 2x angle at circumference"]
K -->|No| M["Angles in the same segment are equal"]
Worked Example: Putting It All Together
Let’s solve a typical GCSE problem.
Problem: In the diagram, O is the centre. AB is a diameter. Angle BAC = 32°. Find angle OCA.
(For this example, imagine a diagram where A, B, C are on the circle, with centre O. AB is a diameter. Line AC is drawn, and line OC is a radius.)
Step-by-Step Solution:
Identify known theorems. AB is a diameter, so triangle ABC is in a semicircle. Therefore, angle ACB = 90° (Theorem 3).
Use triangle rules. In triangle ABC, we now know angles at A (32°) and C (90°). So angle ABC = 180 - 90 - 32 = 58°.
- Look for other theorems. Triangle OBC is isosceles because OB and OC are radii. Therefore, angle OBC = angle OCB.
We know angle OBC is the same as angle ABC, which is 58°. So angle OCB = 58°.
- Calculate the target angle. Angles around point C on line AC are: angle OCA + angle OCB = angle ACB.
So, angle OCA + 58° = 90°.
Therefore, angle OCA = 32°.
Notice how we used Theorem 3 (angle in a semicircle) and basic triangle geometry together.
Top Tips for Exam Success
Label everything: As you find an angle, write it on the diagram.
Look for isosceles triangles: Radii are equal, so triangles with two radii as sides are isosceles.
Combine theorems: Many questions require using 2 or 3 theorems in one problem.
Practise with past papers: Each exam board (AQA, Edexcel, OCR) has its own style. Get used to their question formats.
- Justify your answer: In written questions, always state the theorem you are using (e.g., “Angles in the same segment are equal”).
Common Pitfalls to Avoid
Misidentifying the angle at the centre: Ensure both angles are subtended by the exact same arc.
Forgetting the cyclic quadrilateral rule: Check all four vertices are on the circle before using Theorem 4.
- Assuming lines are parallel or perpendicular: Only use the theorems listed. Don’t make assumptions unless a diagram is marked with the right angle symbol or arrows for parallel lines.
Final Thoughts
Circle theorems are like a toolkit. With practice, you’ll quickly learn which tool (theorem) to use for each part of a problem. Start by learning each theorem individually, then practise combined problems. Remember to use the flowchart strategy to organise your thinking. Good luck with your GCSE revision!
- Need more practice? Search for GCSE circle theorem questions on your exam board's website or via BBC Bitesize Maths.