Circle Theorems Formula Sheet with Examples
GCSE circle theorem examples with formulae, angle facts, worked questions, common mistakes and examiner guidance.
Circle theorem questions reward clear reasoning. The formula or theorem is only part of the answer: you also need to explain why it applies. Students often know several angle facts separately, but the exam skill is spotting which fact matches the diagram and then linking it to ordinary angle rules. That is why clear reasons, tidy annotations and accurate angle labels matter in every worked solution, especially when multiple theorems combine in harder exam questions with unfamiliar diagrams and hidden angles.
This guide is written for GCSE Maths students who want more than a formula list. It shows the method, the common traps and the exam thinking behind each worked example.
For the formula background, use GCSE Maths Circle Formula Sheet. For one-to-one support, see GCSE Maths tutoring or contact Sophie.
The key formula or method
Angle at centre = 2 x angle at circumference. Opposite angles in a cyclic quadrilateral add to 180 degrees. Radius and tangent meet at 90 degrees.
Before using any formula, write down what each letter represents and check the units. Most GCSE mistakes come from rushing the setup, not from the final arithmetic.
Worked examples
Angle at the centre
Method: If an angle at the circumference is 38 degrees and both angles stand on the same arc, the angle at the centre is 76 degrees.
What the examiner is looking for: The examiner wants the theorem named or clearly described.
After completing this example, cover the solution and try to write the key first step from memory. That first decision is usually what separates a confident exam answer from a guessed one.
Then ask yourself why this method was chosen. Was it a particular word in the question, a diagram feature, a formula structure, a unit, or the type of unknown? That explanation is important because it helps you recognise the same idea when the numbers and context change.
Finally, rewrite the answer in a cleaner exam style: formula first, substitution second, calculation third, final answer with units or suitable rounding last. This builds the habit of showing enough working for method marks.
Angle in a semicircle
Method: If a triangle is drawn inside a semicircle with the diameter as one side, the angle at the circumference is 90 degrees.
What the examiner is looking for: The key reason is angle in a semicircle is a right angle.
After completing this example, cover the solution and try to write the key first step from memory. That first decision is usually what separates a confident exam answer from a guessed one.
Then ask yourself why this method was chosen. Was it a particular word in the question, a diagram feature, a formula structure, a unit, or the type of unknown? That explanation is important because it helps you recognise the same idea when the numbers and context change.
Finally, rewrite the answer in a cleaner exam style: formula first, substitution second, calculation third, final answer with units or suitable rounding last. This builds the habit of showing enough working for method marks.
Cyclic quadrilateral
Method: Opposite angles in a cyclic quadrilateral add to 180 degrees. If one angle is 112 degrees, the opposite angle is 68 degrees.
What the examiner is looking for: Show the subtraction from 180 for the method mark.
After completing this example, cover the solution and try to write the key first step from memory. That first decision is usually what separates a confident exam answer from a guessed one.
Then ask yourself why this method was chosen. Was it a particular word in the question, a diagram feature, a formula structure, a unit, or the type of unknown? That explanation is important because it helps you recognise the same idea when the numbers and context change.
Finally, rewrite the answer in a cleaner exam style: formula first, substitution second, calculation third, final answer with units or suitable rounding last. This builds the habit of showing enough working for method marks.
Tangent and radius
Method: A radius to a tangent meets it at 90 degrees. Use this to create right-angled triangles in harder questions.
What the examiner is looking for: The examiner checks whether you identify the right angle at the point of contact.
After completing this example, cover the solution and try to write the key first step from memory. That first decision is usually what separates a confident exam answer from a guessed one.
Then ask yourself why this method was chosen. Was it a particular word in the question, a diagram feature, a formula structure, a unit, or the type of unknown? That explanation is important because it helps you recognise the same idea when the numbers and context change.
Finally, rewrite the answer in a cleaner exam style: formula first, substitution second, calculation third, final answer with units or suitable rounding last. This builds the habit of showing enough working for method marks.
Alternate segment theorem
Method: The angle between a tangent and chord equals the angle in the alternate segment.
What the examiner is looking for: This theorem is often missed because the matching angle is not adjacent to the tangent.
After completing this example, cover the solution and try to write the key first step from memory. That first decision is usually what separates a confident exam answer from a guessed one.
Then ask yourself why this method was chosen. Was it a particular word in the question, a diagram feature, a formula structure, a unit, or the type of unknown? That explanation is important because it helps you recognise the same idea when the numbers and context change.
Finally, rewrite the answer in a cleaner exam style: formula first, substitution second, calculation third, final answer with units or suitable rounding last. This builds the habit of showing enough working for method marks.
Combining circle theorem and triangle facts
Method: After finding one circle angle, use angles in a triangle or angles on a straight line to complete the problem.
What the examiner is looking for: Multi-step reasoning needs each angle fact shown clearly.
After completing this example, cover the solution and try to write the key first step from memory. That first decision is usually what separates a confident exam answer from a guessed one.
Then ask yourself why this method was chosen. Was it a particular word in the question, a diagram feature, a formula structure, a unit, or the type of unknown? That explanation is important because it helps you recognise the same idea when the numbers and context change.
Finally, rewrite the answer in a cleaner exam style: formula first, substitution second, calculation third, final answer with units or suitable rounding last. This builds the habit of showing enough working for method marks.
Common mistakes
- Starting with numbers before deciding which method is needed.
- Missing brackets when substituting negative values or multi-part expressions.
- Rounding too early and carrying an inaccurate answer into later steps.
- Writing a final answer without units when the question needs them.
- Not showing enough working for method marks.
- Using a formula correctly but with values from the wrong part of the diagram or question.
How to revise this topic
Choose two examples from this page and redo them without looking. Then change one number in each question and solve the new version. This is a simple way to check whether you understand the method rather than remembering the answer.
Next, find three mixed past-paper questions where the topic is not named in the title. Mixed practice matters because exams rarely tell you which formula to use. The aim is to recognise the trigger in the question.
When marking, compare your working with the mark scheme. Look for method marks, not just final answers. If your final answer is wrong but your setup is correct, that is a different problem from choosing the wrong method at the start.
Keep a correction log with short labels such as formula choice, substitution, calculator input, rearranging, units or rounding. These labels make revision more targeted and less overwhelming.
What strong exam working looks like
Strong working is not about writing a long essay. It is about making the mathematical route clear. A good answer normally begins with the formula, theorem or equation that connects the information in the question. Then it shows substitution, one or two sensible calculation steps, and a final answer that matches the question.
This matters because GCSE mark schemes often reward method. A student can sometimes recover marks even if the arithmetic goes wrong, but only if the examiner can see a valid method. If the answer jumps straight from the question to a calculator result, there may be very little evidence to reward.
Good notation also reduces mistakes. Keep equals signs lined up, avoid squeezing several steps onto one line, and use brackets around negative numbers or grouped expressions. These small habits are especially useful when questions involve formulae, angles, fractions, square roots or rearranging.
If the question is worded as a real-life problem, finish by checking that the answer is realistic. A negative length, an angle bigger than the diagram suggests, a speed with the wrong unit, or a pressure answer with no area unit should all make you pause before moving on.
How parents can support this practice
Parents do not need to reteach the whole topic to be helpful. A useful role is to ask the student to explain the first step, the formula choice and the final unit. If the student can explain those three things clearly, they usually understand much more than just the answer.
It is also helpful to separate confidence from accuracy. A student might understand the method but make a calculator slip, or they might get the answer right without understanding why. The correction should match the real issue, otherwise revision can become frustrating and unfocused.
Short, regular sessions usually work better than one long session. Ten minutes spent redoing one example carefully, checking the working and writing one correction can be more valuable than rushing through a full worksheet without review.
If the same type of mistake appears several times, that is a good sign the topic needs more guided practice. This is where tutoring can help, because the lesson can slow down the exact step that is causing the problem.
A simple three-session revision plan
In the first session, focus only on understanding the examples. Read each question, cover the method, and predict the first line of working. Do not worry about speed at this point. The aim is to make the decision process feel clear.
In the second session, practise accuracy. Redo the examples without looking, then check every substitution, bracket, unit and rounding choice. If you use a calculator, type the calculation again to check whether the original input was reliable.
In the third session, move to mixed questions. Choose questions from different worksheets or past papers so the topic is not obvious from the page title. This helps you practise recognising the method under exam conditions.
At the end of the third session, write a short summary: the formula or method, the trigger that tells you to use it, the most common mistake, and one example question to redo next week. This turns the topic into a reusable revision note rather than a one-off exercise.
When to move on
Move on when you can complete a straightforward question, a question with one extra step, and a mixed exam-style question without needing the answer in front of you. If you can only do the straightforward version, the topic is not secure yet.
It is completely normal for this to take more than one attempt. GCSE Maths topics become secure through repeated retrieval, not through reading a perfect worked solution once. The goal is steady independence: fewer prompts, clearer working and more confidence with unfamiliar wording.
If you are aiming for higher marks, add a final challenge: explain the method to someone else or write your own question that uses the same idea. Creating a question forces you to understand the structure of the method much more deeply.
How this links to wider GCSE Maths revision
Worked examples should sit between formula revision and full past papers. Formula revision helps you remember the tools. Worked examples show the tools in use. Past papers then test whether you can choose between several possible tools under time pressure.
This middle stage is often the one students skip. They either reread notes or jump straight into a paper. That can work for topics that already feel secure, but it is less effective for topics where the method still feels fragile. Worked examples give you a bridge from understanding to independence.
For best results, keep the examples close to your formula sheet. When you revise a formula, immediately pair it with one worked example and one exam question. That makes the formula meaningful and reduces the chance of forgetting it when the question is worded differently.
If you are preparing for mocks or final exams, revisit this page after a week and again after a month. Spaced repetition is especially useful in Maths because it shows whether a method has genuinely stuck or only felt familiar on the day you first learnt it.
A useful final routine is to keep one page in your revision folder called methods to remember. Add the example title, the first line of working, and the mistake to avoid. This gives you a quick, personal checklist before tests, mocks and independent revision.
Related formula and tutoring guides
Use these related pages to connect the worked examples with formula revision and more personalised support.
Worked example guides
Once you know the formulae, these worked-example guides show how to turn them into method marks in exam-style questions.
- How to Use the Quadratic Formula: GCSE Worked Examples
- Sine Rule and Cosine Rule GCSE Questions Explained
- Circle Theorems Formula Sheet with Examples
- Speed, Density and Pressure Formula Questions
- How to Rearrange Formulae for GCSE Maths
FAQs
How many worked examples should I practise?
Start with five carefully marked examples, then move to mixed exam questions. Quality of review matters more than the number of questions completed.
Should I memorise the formula first?
You should know the formula, but the bigger skill is recognising when to use it and substituting accurately.
Why do I lose marks even when my answer is close?
Marks can be lost through missing working, early rounding, wrong units, poor notation or choosing the right formula for the wrong values.
How should I correct mistakes?
Write the exact reason for the mistake, then redo a similar question a few days later.
Can tutoring help with worked examples?
Yes. Tutoring can slow down the method, identify gaps and help students understand what examiners reward.
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