A triangle is drawn in a semicircle. What is the angle x?
The angle in a semicircle is always a right angle.
The angle in a semicircle is 90°.
Circle Theorems
Circle theorems describe relationships between angles in a circle. They are essential for solving higher GCSE Maths geometry problems and often combine with angle rules and trigonometry.
Overview
Circle theorems are special angle rules inside circles.
In exam questions, you usually need to identify the correct theorem first, then combine it with basic angle facts such as angles on a straight line, angles around a point, or angles in a triangle.
Spot the theorem → write the angle fact → solve step by step
Most students do not lose marks because the maths is hard. They lose marks because they do not recognise which circle theorem is being used.
What you should understand after this topic
- Know the main GCSE circle theorems
- Recognise each theorem from a diagram
- Combine theorem facts with angle rules
- Explain reasoning clearly in solutions
- Avoid common exam mistakes
Key Definitions
Radius
A line from the centre of the circle to the circumference.
Diameter
A straight line through the centre joining two points on the circle.
Chord
A straight line joining two points on the circumference.
Tangent
A straight line that touches the circle at exactly one point.
Arc
A curved part of the circumference.
Centre
The middle point of the circle.
Alternate Segment
The opposite side of a chord used in the tangent-chord theorem.
Cyclic Quadrilateral
A quadrilateral with all 4 vertices on the circumference.
Key Rules
Key Theorems
Theorem 1
Angle at the centre is twice the angle at the circumference
Angles standing on the same arc are linked by a factor of 2.
Theorem 2
Angles in the same segment are equal
If two angles stand on the same chord, they are equal.
Theorem 3
Angle in a semicircle is \(90^\circ\)
If a triangle is drawn on a diameter, the angle at the circumference is a right angle.
Theorem 4
Opposite angles in a cyclic quadrilateral add to \(180^\circ\)
This is one of the most common exam theorems.
Theorem 5
Radius is perpendicular to a tangent
The angle between a radius and a tangent is always \(90^\circ\).
Theorem 6
Two tangents from the same point are equal
If two tangents meet outside a circle, the tangent lengths are the same.
Theorem 7
Angle between tangent and chord equals angle in the alternate segment
This is often called the alternate segment theorem.
Theorem 8
Perpendicular from centre to chord bisects the chord
If a line from the centre meets a chord at \(90^\circ\), it cuts the chord into two equal parts.
How to Solve
How to approach a circle theorem question
Circle theorem questions are about recognising patterns. Before calculating anything, study the diagram carefully and identify the theorem.
Look for key features: diameter, chord, tangent or centre.
Match the diagram to a known theorem.
State the theorem clearly before calculating.
Use angle rules to complete the solution.
Exam tip: Most marks come from identifying and stating the correct theorem.
1. Angle at the centre
Angle at centre = 2 × angle at circumference
Both angles must stand on the same arc.
Example: \(35^\circ → 70^\circ\).
2. Same segment theorem
Angles in the same segment are equal.
Exam tip: Look for angles touching the same arc.
3. Angle in a semicircle
Angle in a semicircle = \(90^\circ\)
Occurs when a triangle is drawn on a diameter.
4. Cyclic quadrilateral
Opposite angles add to \(180^\circ\)
Example: \(112^\circ + 68^\circ = 180^\circ\).
5. Radius and tangent
Radius ⟂ Tangent
The angle at the point of contact is always \(90^\circ\).
6. Alternate segment theorem
Angle between tangent and chord = angle in opposite segment.
7. Tangents from the same point
Tangents from the same external point are equal in length.
8. Centre to chord
A perpendicular from the centre to a chord bisects the chord.
Combining theorems
Most exam questions require combining multiple theorems with angle rules.
You will often combine this with angles and trigonometry.
Exam tip: Work step by step and justify each angle.
Example Questions
Edexcel
Exam-style questions focusing on angles at the centre and angles in a semicircle.
Edexcel
An angle at the circumference is 32°.
Find the angle at the centre standing on the same arc.
Edexcel
A triangle is drawn in a semicircle.
What is the angle opposite the diameter?
AQA
Exam-style questions focusing on cyclic quadrilaterals and tangent-radius angle facts.
AQA
One angle in a cyclic quadrilateral is 107°.
Find the opposite angle x.
AQA
A tangent touches the circle at point T. A radius is drawn to T.
Find the angle x between the radius and the tangent.
OCR
Exam-style questions focusing on angles in the same segment and mixed semicircle reasoning.
OCR
Two angles stand on the same chord.
Explain why the two angles are equal.
OCR
A triangle is drawn in a semicircle. Another angle in the triangle is 41°.
Find the third angle x.
OCR
The angle at the centre of a circle is 126°.
Find the angle at the circumference standing on the same arc.
Exam Checklist
Step 1
Look for clues such as diameter, tangent, chord or centre.
Step 2
Identify the circle theorem before doing any calculation.
Step 3
Write the theorem fact clearly on the diagram or in working.
Step 4
Use triangle, straight-line or quadrilateral angle rules to finish.
Most common exam mistakes
Centre confusion
Using equal instead of double for centre and circumference angles.
Semicircle miss
Not spotting that a line is a diameter.
Cyclic miss
Forgetting opposite angles in a cyclic quadrilateral sum to \(180^\circ\).
Tangent miss
Forgetting the right angle between radius and tangent.
Common Mistakes
These are common mistakes students make when applying circle theorems in GCSE Maths.
Using the wrong theorem
Incorrect
A student applies a theorem without carefully analysing the diagram.
Correct
Always identify key features first (centre, tangent, chord, cyclic quadrilateral). Choose the correct theorem based on what is shown in the diagram.
Confusing centre and circumference angles
Incorrect
A student says the angle at the centre is equal to the angle at the circumference.
Correct
The angle at the centre is twice the angle at the circumference when they stand on the same arc. This is a very common exam mistake.
Missing a cyclic quadrilateral
Incorrect
A student does not recognise that four points lie on a circle.
Correct
If all four vertices lie on the circle, the shape is cyclic. Opposite angles in a cyclic quadrilateral always add up to \(180^\circ\).
Forgetting the tangent rule
Incorrect
A student does not recognise a right angle between a radius and a tangent.
Correct
A radius meets a tangent at \(90^\circ\). This is often the first step in solving circle theorem problems.
Not combining theorems with basic angle facts
Incorrect
A student stops after applying one theorem and does not continue.
Correct
Circle theorems are usually combined with standard angle rules (e.g. angles in a triangle or on a straight line). Continue solving until the final value is found.
Try It Yourself
Practise applying circle theorems to calculate missing angles.
Foundation
Foundation Practice
Use the main circle theorem facts for angles in semicircles, at the centre and in cyclic quadrilaterals.
The angle at the circumference is 35°. Find the angle at the centre standing on the same arc.
The angle at the centre is twice the angle at the circumference.
2 × 35° = 70°.
A tangent touches a circle at T. A radius is drawn to T. What is the angle x?
A tangent is perpendicular to the radius at the point of contact.
The radius and tangent meet at 90°.
One angle in a cyclic quadrilateral is 110°. Find the opposite angle x.
Opposite angles in a cyclic quadrilateral add to 180°.
180° − 110° = 70°.
Two angles stand on the same chord. What can you say about angles a and b?
Angles in the same segment are equal.
Angles standing on the same chord are equal.
The angle at the centre is 124°. Find the angle at the circumference standing on the same arc.
The angle at the circumference is half the angle at the centre.
124° ÷ 2 = 62°.
Which theorem says that the angle between a tangent and a radius is 90°?
It involves a radius and tangent.
The tangent-radius theorem says the radius and tangent meet at 90°.
A triangle is drawn in a semicircle. One base angle is 38°. Find the other base angle x.
Angles in a triangle add to 180°.
180° − 90° − 38° = 52°.
Opposite angles in a cyclic quadrilateral:
Use the cyclic quadrilateral theorem.
Opposite angles in a cyclic quadrilateral add to 180°.
Angles a and b are in the same segment. If a = 47°, find b.
Angles in the same segment are equal.
b = 47°.
Higher
Higher Practice
Solve multi-step circle theorem problems, including tangent, cyclic quadrilateral and alternate segment reasoning.
The angle between the tangent and chord is 52°. Find x using the alternate segment theorem.
The angle between a tangent and chord equals the angle in the opposite segment.
By the alternate segment theorem, x = 52°.
A cyclic quadrilateral has angles 82°, 98°, 115° and x. Find x.
Opposite angles add to 180°.
x is opposite 115°, so x = 180° − 115° = 65°.
The angle at the circumference is 41°. Find the reflex angle at the centre standing on the same arc.
Find the smaller centre angle first, then subtract from 360°.
Smaller centre angle = 2 × 41° = 82°. Reflex angle = 360° − 82° = 278°.
A tangent and chord form an angle of 68°. What is the angle in the alternate segment?
Use the alternate segment theorem.
The alternate segment angle is equal to the tangent-chord angle, so it is 68°.
In a cyclic quadrilateral, one angle is 3x and the opposite angle is x + 40. Find x.
Opposite angles add to 180°.
3x + x + 40 = 180, so 4x = 140 and x = 35.
A radius meets a tangent at T. Another angle around T is 37°. What is the remaining angle x?
The radius and tangent make 90° altogether.
x = 90° − 37° = 53°.
A triangle is drawn in a semicircle. The two smaller angles are x and 2x. Find x.
The angle in the semicircle is 90°.
x + 2x + 90 = 180, so 3x = 90 and x = 30°.
Which theorem is being used when the angle at the centre is twice the angle at the circumference?
It directly compares centre and circumference angles.
This is the angle at the centre theorem.
Two angles in the same segment are labelled 2x + 10 and 4x − 30. Find x.
Angles in the same segment are equal.
2x + 10 = 4x − 30, so 40 = 2x and x = 20.
A cyclic quadrilateral has opposite angles labelled 5x and 2x + 40. Find x.
Opposite angles add to 180°.
5x + 2x + 40 = 180, so 7x = 140 and x = 20.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is the angle at the centre rule?
The angle at the centre is twice the angle at the circumference.
What happens in a semicircle?
The angle is always 90 degrees.
What are cyclic quadrilaterals?
Opposite angles add up to 180 degrees.