Circle theorems are facts about angles in circles. The angle at the centre is twice the angle at the circumference, the angle in a semicircle is 90°, angles in the same segment are equal, and opposite angles of a cyclic quadrilateral add up to 180°. A tangent meets a radius at 90°, two tangents from a point are equal, and the angle between a tangent and a chord equals the angle in the alternate segment.
The circle theorems
- The angle at the centre is twice the angle at the circumference
- The angle in a semicircle is 90°
- Angles in the same segment are equal
- Opposite angles of a cyclic quadrilateral add up to 180°
- A tangent is perpendicular to the radius at the point of contact
- Tangents from an external point are equal in length
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment
Angle at the centre and the semicircle
When an arc is “seen” from the centre and from a point on the circumference, the angle at the centre is twice the angle at the circumference. A special case is a diameter: the angle at the centre is 180°, so the angle in a semicircle is always 90°. Any triangle drawn with one side as a diameter is right-angled.
Same segment and cyclic quadrilaterals
Angles standing on the same arc, on the same side of the chord, are equal. A cyclic quadrilateral has all four vertices on a circle, and its opposite angles add up to 180°. If one angle is 72°, the opposite angle is 108°.
Tangent theorems
A tangent touches a circle at exactly one point and is perpendicular to the radius there. Two tangents drawn from the same external point are equal in length, which often creates an isosceles triangle or a kite. In the quadrilateral formed by two tangents and two radii, the two right angles mean the other two angles add up to 180°.
The alternate segment theorem
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment, the angle subtended by the chord on the opposite side. This is often the hardest theorem to spot; look for a tangent meeting a chord at the point of contact.
Giving reasons
Circle theorem questions nearly always say “give reasons for your answer”. Quote each theorem in full, such as “opposite angles in a cyclic quadrilateral add up to 180°”, and include ordinary angle facts as well. Abbreviated or vague reasons may not earn the marks.
Worked examples
Example 1. The angle at the circumference standing on an arc AB is 38°. Find the angle at the centre on the same arc.
- The angle at the centre is twice the angle at the circumference.
- 2 × 38.
Answer: 76°.
Example 2. AB is a diameter of a circle and C is on the circumference. Angle CAB is 34°. Find angle CBA.
- The angle in a semicircle is 90°, so angle ACB = 90°.
- Angles in a triangle: 180 − 90 − 34.
Answer: 56°.
Example 3. ABCD is a cyclic quadrilateral with angle ABC = 72°. Find angle ADC.
- Opposite angles in a cyclic quadrilateral add up to 180°.
- 180 − 72.
Answer: 108°.
Example 4. PA and PB are tangents to a circle with centre O. Angle APB = 50°. Find angle AOB.
- Tangents meet radii at 90°, so angles OAP and OBP are 90°.
- Angles in quadrilateral OAPB: 360 − 90 − 90 − 50.
Answer: 130°.
Example 5. A tangent at A makes an angle of 63° with the chord AB. C is on the circumference in the alternate segment. Find angle ACB.
- Alternate segment theorem: the angle between the tangent and chord equals the angle in the alternate segment.
Answer: 63°.
Common mistakes
- Halving instead of doubling. The angle at the centre is twice the angle at the circumference.
- Using the cyclic quadrilateral rule for adjacent angles. It is opposite angles that add to 180°.
- Forgetting the right angle at a tangent. Tangent and radius meet at 90°.
- Vague reasons. Write each theorem out in full.
- Using the same-segment rule across the chord. Both angles must be on the same side of the chord.
How circle theorems are examined
Circle theorems appear only on Higher papers, for both GCSE and iGCSE, and almost every question asks for reasons. Questions ask you to:
- find a missing angle using one or more theorems
- give a reason for each step
- combine circle theorems with triangle and parallel line facts
- prove a circle theorem (Higher, occasionally)
The circles and circle theorems worksheet includes worked answers, and the GCSE geometry revision engine gives more practice.
Circle theorems build on angle rules. At A-Level, circles are studied with coordinates in tangents and chords.
Frequently asked questions
How many circle theorems are there at GCSE?
Usually seven or eight are learned, including the angle at the centre, the semicircle, same segment, cyclic quadrilaterals, the tangent theorems and the alternate segment theorem.
What is the angle in a semicircle?
90°. Any angle drawn from a diameter to the circumference is a right angle.
What is the alternate segment theorem?
The angle between a tangent and a chord equals the angle in the alternate segment.