This GCSE Maths circle theorems worksheet with answers is designed for focused GCSE revision. Circle theorem questions need answers and reasons. Students should practise naming the theorem, then using ordinary angle facts to finish the question.
Use it as a short practice task before moving into mixed exam questions. If you need more structured help, I also offer GCSE Maths tutoring and parent-friendly GCSE revision support.
Before you start
The level for this worksheet is Higher. Work through the questions without checking the answers first. The point is not only to get the final answer, but to practise the method clearly enough that you could repeat it in a test.
Keep your working visible. GCSE Maths mark schemes often reward method, so a clear first line can still matter even if the arithmetic goes wrong later.
Skills this worksheet checks
This worksheet checks whether you can recognise common circle theorems question types, choose a sensible first step, and keep your working organised. Those three habits matter more than racing through the list.
For each question, write one line that explains the method before you calculate. That might be a formula, a common denominator, an equation, a route through a diagram, or the probability operation you are using. This makes the practice closer to the way marks are awarded in GCSE exams.
Worksheet questions
- Angle at the circumference is 34 degrees. Find the angle at the centre on the same arc.
- An angle in a semicircle stands on the diameter. Find the angle.
- A cyclic quadrilateral has one angle 112 degrees. Find the opposite angle.
- A radius meets a tangent at the point of contact. Find the angle between them.
- Two angles at the circumference stand on the same chord. One is 47 degrees. Find the other.
- The angle between a tangent and chord is 62 degrees. Find the angle in the alternate segment.
- A triangle has vertices on a circle. One side is a diameter. The other acute angle is 38 degrees. Find the third angle.
- A cyclic quadrilateral has adjacent angles 85 and 97 degrees. Find the other two if they are opposite these respectively.
- The centre angle is 146 degrees. Find the circumference angle on the same arc.
- A tangent creates a right angle with a radius. Another angle in the triangle is 36 degrees. Find the third angle.
Answers and method checks
| # | Question | Method | Answer |
|---|---|---|---|
| 1 | Angle at the circumference is 34 degrees. Find the angle at the centre on the same arc. | The angle at the centre is twice the angle at the circumference. | 68 degrees |
| 2 | An angle in a semicircle stands on the diameter. Find the angle. | Angle in a semicircle is a right angle. | 90 degrees |
| 3 | A cyclic quadrilateral has one angle 112 degrees. Find the opposite angle. | Opposite angles in a cyclic quadrilateral add to 180 degrees. | 68 degrees |
| 4 | A radius meets a tangent at the point of contact. Find the angle between them. | Radius and tangent are perpendicular. | 90 degrees |
| 5 | Two angles at the circumference stand on the same chord. One is 47 degrees. Find the other. | Angles in the same segment are equal. | 47 degrees |
| 6 | The angle between a tangent and chord is 62 degrees. Find the angle in the alternate segment. | Use the alternate segment theorem. | 62 degrees |
| 7 | A triangle has vertices on a circle. One side is a diameter. The other acute angle is 38 degrees. Find the third angle. | The diameter angle is 90 degrees, then use triangle angles. | 52 degrees |
| 8 | A cyclic quadrilateral has adjacent angles 85 and 97 degrees. Find the other two if they are opposite these respectively. | Subtract each given angle from 180 degrees. | 95 and 83 degrees |
| 9 | The centre angle is 146 degrees. Find the circumference angle on the same arc. | The circumference angle is half the centre angle. | 73 degrees |
| 10 | A tangent creates a right angle with a radius. Another angle in the triangle is 36 degrees. Find the third angle. | Use angles in a triangle after the tangent-radius fact. | 54 degrees |
How to mark this worksheet
Mark each question with more detail than a tick or cross. If the answer is wrong, label the error: method choice, arithmetic, notation, units, signs, calculator input, or not reading the question carefully.
For circle theorems, it is especially useful to redo the questions you missed after a short break. A topic can feel familiar immediately after reading the answer, but the real test is whether you can start the method independently later.
A good correction is specific. Instead of writing 'revise this', write the exact fix: choose a common denominator, show the multiplier, update the probability after the first choice, label the radius, or subtract the smaller coordinate from the larger one. Specific corrections make the next practice session much easier to plan.
Common mistakes
- Giving the angle without the theorem reason.
- Doubling when the question needs halving.
- Using cyclic quadrilateral facts on a shape that is not cyclic.
- Missing ordinary triangle or straight-line angle facts after the theorem.
Extension practice
After marking the worksheet, choose three questions and change one number in each. Keep the structure the same, then solve your new versions. This is a quick way to check whether you understand the method rather than remembering the original answer.
If the changed question becomes much harder, that is useful information. It usually means the method is not fully secure yet, or that a small change has introduced a new skill such as negative numbers, fractions, unit conversion, exact form, or a second step.
Students aiming for higher grades should also write one short explanation question. For example: explain why this vector is parallel, explain why these two routes must be added, explain why this theorem applies, or explain why the original amount is not the same as the sale amount. These explanations build the reasoning marks that many students miss.
What to practise next
Once you can complete this worksheet accurately, move to mixed questions where the topic is not written in the title. That is closer to the exam, because you have to recognise the method as well as carry it out.
You can also use the Maths practice papers hub to connect topic practice with broader exam-style work. If the same mistake keeps appearing, bring that exact question to a lesson or send it to me so we can work out what is blocking the method.
