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Circle theorems practice questions with answers

Higher · Grade 6 · Non-calculator

A second set of circle theorems questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

Angle at the centre is 70 degrees. Find the angle at the circumference.

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Answer: 35

  1. Circumference angle is half centre angle.
  2. 70 / 2 = 35.
  • Method (1): Circumference angle is half centre angle.
  • Accuracy (1): 70 / 2 = 35.

Question 2 Easy

Angle at the centre is 90 degrees. Find the angle at the circumference.

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Answer: 45

  1. Circumference angle is half centre angle.
  2. 90 / 2 = 45.
  • Method (1): Circumference angle is half centre angle.
  • Accuracy (1): 90 / 2 = 45.

Question 3 Medium

Angle at the circumference is 25 degrees. Find the angle at the centre.

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Answer: 50

  1. Centre angle is double circumference angle.
  2. 25 × 2 = 50.
  • Method (1): Centre angle is double circumference angle.
  • Accuracy (1): 25 × 2 = 50.

Question 4 Medium

Angle at the circumference is 48 degrees. Find the angle at the centre.

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Answer: 96

  1. Centre angle is double circumference angle.
  2. 48 × 2 = 96.
  • Method (1): Centre angle is double circumference angle.
  • Accuracy (1): 48 × 2 = 96.

Question 5 Hard

One angle of a cyclic quadrilateral is 80 degrees. Find the opposite angle.

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Answer: 100

  1. Opposite angles add to 180.
  2. 180 - 80 = 100.
  • Method (1): Opposite angles add to 180.
  • Accuracy (1): 180 - 80 = 100.

Question 6 Hard

One angle of a cyclic quadrilateral is 65 degrees. Find the opposite angle.

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Answer: 115

  1. Opposite angles add to 180.
  2. 180 - 65 = 115.
  • Method (1): Opposite angles add to 180.
  • Accuracy (1): 180 - 65 = 115.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Angle at the centre is 140 degrees. Find the angle at the circumference on the same arc.

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Answer: 70

  1. The angle at the circumference is half the angle at the centre on the same arc.
  2. Divide by 2: 140 ÷ 2 = 70 degrees.
  • Method (1): Halves the centre angle: 140 ÷ 2
  • Accuracy (1): 70

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

Half of a centre angle is the angle at the

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Circumference angle is half the centre angle on the same arc. Circumference angle is half the centre angle on the same arc.

Check 2

Centre 80 degrees gives a circumference angle of

  1. 70 degrees
  2. 40 degrees
  3. 80 degrees
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40 degrees 80 halved is 40 degrees.

Check 3

Centre 100 degrees gives a circumference angle of

  1. 50 degrees
  2. 40 degrees
  3. 80 degrees
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50 degrees 100 halved is 50 degrees.

Check 4

Circumference 35 degrees gives a centre angle of

  1. 70 degrees
  2. 40 degrees
  3. 50 degrees
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70 degrees 35 doubled is 70 degrees.

Check 5

Circumference 40 degrees gives a centre angle of

  1. 50 degrees
  2. 80 degrees
  3. 40 degrees
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80 degrees 40 doubled is 80 degrees.

Check 6

The angle in a semicircle is

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A diameter subtends a right angle at the circumference. A diameter subtends a right angle at the circumference.

Check 7

The angle between a tangent and the radius is

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A radius meets a tangent perpendicular at 90 degrees. A radius meets a tangent perpendicular at 90 degrees.

Check 8

Opposite angles of a cyclic quadrilateral add to

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Opposite vertices on a circle have angles summing to 180. Opposite vertices on a circle have angles summing to 180.

Check 9

One cyclic angle 70 degrees leaves the opposite angle

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180 - 70 = 110 degrees. 180 - 70 = 110 degrees.

Check 10

One cyclic angle 80 degrees leaves the opposite angle

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180 - 80 = 100 degrees. 180 - 80 = 100 degrees.

Check 11

Why not double a centre angle to get the circumference angle?

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The angle at circumference is half, not double. The angle at circumference is half, not double.

Check 12

Same arc means the centre angle and the circumference angle are

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Both angles stand on the exact same arc of the circle. Both angles stand on the exact same arc of the circle.

Check 13

A radius and a tangent meet at

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Tangents are perpendicular to the radius at the point of contact. Tangents are perpendicular to the radius at the point of contact.

Check 14

Semicircle theorem needs which side as diameter?

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The hypotenuse of the triangle must be the circle diameter. The hypotenuse of the triangle must be the circle diameter.

Check 15

180 minus a cyclic angle gives the

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The opposite interior angle of the cyclic quad. The opposite interior angle of the cyclic quad.

Keep going with Circle theorems

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