Circle theorems are a crucial part of the GCSE Maths curriculum across AQA, Edexcel, and OCR. They are frequently tested in higher-tier papers and often carry significant marks when combined with algebra and geometric reasoning. In this guide, we break down each key circle theorem with diagrams, tips, and practice questions to ensure you’re fully prepared.

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🔹 What Are Circle Theorems?

Circle theorems are rules about angles and lengths in and around circles. These theorems help you solve problems involving tangents, chords, sectors, and cyclic quadrilaterals.

✅ Why They Matter:

• Appear in both short and long exam questions
• Often combined with other topics like algebra and angles in polygons
• Require detailed reasoning and labelled diagrams

🔸 Key Circle Theorems (with Examples)

1. Angle at the Centre = 2 × Angle at the Circumference

If two angles subtend the same arc:

Angle at centre = 2 × angle at circumference

2. Angle in a Semicircle = 90°

A triangle drawn from the diameter forms a right angle.

3. Angles in the Same Segment Are Equal

If two angles subtend the same chord, they are equal.

4. Opposite Angles in a Cyclic Quadrilateral Add to 180°

A cyclic quadrilateral is a 4-sided shape with all vertices on the circle.

5. Tangent and Radius Meet at 90°

Tangent ⟂ Radius at the point of contact

6. Tangents from the Same Point Are Equal in Length

Use this when solving for missing lengths or using Pythagoras’ Theorem.

7. Alternate Segment Theorem

The angle between the chord and tangent is equal to the angle in the alternate segment.

🧠 Exam Tips for Circle Theorems

• Always draw and label diagrams clearly
• Include the theorem name in your working
• Use logical steps and link statements
• Write in full sentences when reasoning
• Combine with triangle and angle properties where needed

📝 Practice Questions

1. In a circle, the angle at the centre is 100°. What is the angle at the circumference subtending the same arc?
2. Prove that opposite angles in a cyclic quadrilateral add to 180° using a diagram.
3. Two tangents are drawn from a point outside the circle. One tangent is 5 cm. Find the other.
4. A triangle is inscribed in a semicircle. What type of triangle is it, and why?
5. A tangent and a chord intersect at a point on a circle. The angle between the chord and tangent is 35°. Find the angle in the alternate segment.

✅ Final Thoughts

Circle theorems can appear tricky, but with regular practice and clear diagrams, you can confidently tackle even the most challenging questions. Focus on understanding the logic behind each rule, apply them correctly, and back your solutions with structured reasoning.

For more practice questions, step-by-step video tutorials, and GCSE exam support, visit GCSE Maths Tutor.