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Geometry and measuresHigher

Circle theorems GCSE questions with answers

8 circle theorems questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

Circle theorems are facts about angles in circles, such as the angle at the centre being twice the angle at the circumference. On Higher you use them, along with the facts about tangents, to find missing angles.

Quiz yourself: 8 questions

Worked example 1: circle theorems: angles in a circle

A, B and C are points on a circle with centre O. Work out the size of the angle marked x.

x43°OABC
Not drawn accurately
  1. The angles at O and at C both stand on the arc AB.
  2. The angle at the centre is twice the angle at the circumference.
  3. x = 2 × 43° = 86°.

Answer 86°

01 / Quiz yourselfTry it, then tap to check

Circle theorems questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / Circle theorems: angles in a circleHigher only

A, B and C are points on a circle with centre O. Work out the size of the angle marked x.

x73°OABC
Not drawn accurately
Show the answer and working
  1. The angles at O and at C both stand on the arc AB.
  2. The angle at the centre is twice the angle at the circumference.
  3. x = 2 × 73° = 146°.

Answer 146°

02 / Circle theorems: angles in a circleHigher only

A, B, C and D are points on a circle. Work out the size of the angle marked x.

104°xABCD
Not drawn accurately
Show the answer and working
  1. ABCD is a cyclic quadrilateral: all four corners are on the circle.
  2. Opposite angles of a cyclic quadrilateral add up to 180°.
  3. x = 180° − 104° = 76°.

Answer 76°

03 / Circle theorems: angles in a circleHigher only

A, B and C are points on a circle with centre O. Work out the size of the angle marked x.

x57°OABC
Not drawn accurately
Show the answer and working
  1. The angles at O and at C both stand on the arc AB.
  2. The angle at the centre is twice the angle at the circumference.
  3. x = 2 × 57° = 114°.

Answer 114°

04 / Circle theorems: angles in a circleHigher only

A, B and C are points on a circle with centre O. AB is a diameter. Work out the size of the angle marked x.

34°xOABC
Not drawn accurately
Show the answer and working
  1. AB is a diameter, so angle ACB is an angle in a semicircle. The angle in a semicircle is 90°.
  2. Angles in a triangle add up to 180°: x = 180° − 90° − 34° = 56°.

Answer 56°

05 / Circle theorems: tangentsHigher only

PA and PB are tangents to the circle with centre O. Work out the size of the angle marked x.

46°xOPAB
Not drawn accurately
Show the answer and working
  1. Tangents from a point to a circle are equal in length, so PA = PB and triangle APB is isosceles.
  2. Angle PBA = angle PAB = 46°.
  3. x = 180° − 46° − 46° = 88°.

Answer 88°

06 / Circle theorems: tangentsHigher only

PT is a tangent to the circle at T. O is the centre of the circle. Work out the size of the angle marked x.

x53°OTP
Not drawn accurately
Show the answer and working
  1. OT is a radius and PT is a tangent. The angle between a tangent and a radius is 90°, so angle OTP = 90°.
  2. Angles in a triangle add up to 180°: x = 180° − 90° − 53° = 37°.

Answer 37°

07 / Circle theorems: tangentsHigher only

A, B and C are points on a circle. The line SAT is a tangent to the circle at A. Work out the size of the angle marked x.

40°xABCST
Not drawn accurately
Show the answer and working
  1. Angle BAT is between the tangent and the chord AB.
  2. The angle between a tangent and a chord is equal to the angle in the alternate segment, so x = 40°.

Answer 40°

08 / Circle theorems: tangentsHigher only

A, B and C are points on a circle. The line SAT is a tangent to the circle at A. Work out the size of the angle marked x.

35°xABCST
Not drawn accurately
Show the answer and working
  1. Angle BAT is between the tangent and the chord AB.
  2. The angle between a tangent and a chord is equal to the angle in the alternate segment, so x = 35°.

Answer 35°

02 / What's covered2 kinds of question

Circle theorems at GCSE and where marks go

  • Circle theorems: angles in a circleHigher only
  • Circle theorems: tangentsHigher only

Worked example 2: circle theorems: tangents

A, B and C are points on a circle. The line SAT is a tangent to the circle at A. Work out the size of the angle marked x.

40°xABCST
Not drawn accurately
  1. Angle BAT is between the tangent and the chord AB.
  2. The angle between a tangent and a chord is equal to the angle in the alternate segment, so x = 40°.

Answer 40°

A common mistake

Halving when you should double. The angle at the centre is twice the angle at the circumference from the same arc, so if the angle at the circumference is 35°, the angle at the centre is 70°.

03 / Practise itiPhone and iPad

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