GCSE
Circle Theorems
This might look intimidating — circles covered in lines and angles — but each theorem is actually a simple rule, and there are only seven to learn.
What Your Child Is Learning
Circle theorems are a set of rules about angles and lines inside and around circles. Your child needs to learn seven main theorems and be able to spot which one applies in a diagram.
The key thing to understand is that this topic is less about calculation and more about pattern recognition. When your child sees a circle diagram in an exam, their job is to identify the pattern, name the theorem, and use it to find the missing angle.
Each theorem is a fact that is always true for every circle. Once you know the rules, the questions are actually quite predictable.
How Schools Teach It Now
Schools teach students to learn each theorem, recognise it in a diagram, and — crucially — state the theorem as the reason. At GCSE, both the angle and the reason are usually required for full marks.
Here are the seven key theorems in plain language:
-
1. Angle at centre
The angle at the centre of a circle is twice the angle at the circumference, when both are subtended by the same arc.
-
2. Angle in a semicircle
Any angle inscribed in a semicircle (where the "base" is a diameter) is exactly \(90°\).
-
3. Angles in the same segment
Angles at the circumference in the same segment (on the same side of a chord) are equal.
-
4. Cyclic quadrilateral
Opposite angles in a cyclic quadrilateral (a four-sided shape with all corners on the circle) add up to \(180°\).
-
5. Tangent and radius
A tangent to a circle is perpendicular (\(90°\)) to the radius at the point of contact.
-
6. Two tangents from a point
Two tangents drawn from the same external point are equal in length.
-
7. Alternate segment theorem
The angle between a tangent and a chord equals the angle in the alternate segment.
Parent tip
At Higher tier, your child may be asked to prove theorems as well as apply them. If they can match the diagram to the rule, they are most of the way there.
Common Mistakes to Watch For
Easy marks lost
Not stating the theorem name as the reason. Finding the correct angle is often only half the marks. The exam will usually say "give a reason for your answer" and your child must write the theorem name. "The angle at the centre is twice the angle at the circumference" is worth marks. Just writing "80" with no reason usually loses them.
Classic mistake
Confusing which angle is at the centre vs the circumference. In the "angle at centre = twice angle at circumference" theorem, students sometimes apply it the wrong way round — doubling when they should halve, or vice versa. If the angle at the centre is 80 degrees, the angle at the circumference is 40 degrees (half), not 160 degrees (double).
Watch out
Not spotting the diameter. When a line goes through the centre of the circle, it is a diameter, which means the "angle in a semicircle = 90 degrees" theorem applies. Students sometimes miss that a line is a diameter and overcomplicate the question.
Try It Together
Here is a straightforward example:
The angle at the centre of a circle is 80 degrees. Find the angle at the circumference on the same arc.
- Identify the theorem: this is the "angle at the centre" theorem.
- The rule says: the angle at the centre is twice the angle at the circumference.
- So the angle at the circumference is \(80 \div 2 = 40°\).
- Write the reason: "The angle at the centre is twice the angle at the circumference."
$$\text{Angle at circumference} = \frac{80°}{2} = 40°$$
Conversation starter
Ask your child to draw a circle, mark the centre, and draw two lines from the centre to the edge and two more from the circumference to the same two points on the edge. Then measure the angles — the centre angle should be exactly double the circumference angle. Seeing it with a protractor makes the theorem feel real.
Let Them Practise
Circle theorems are a pattern-spotting exercise. The more diagrams your child sees, the faster they will recognise which theorem to use. It becomes almost automatic with practice.
We have a free interactive game specifically for circle theorem recognition:
Circle Theorem Spotter
Free interactive practice — identify the theorem, find the angle, build confidence.
Practise Now on MaffsGames
MaffsGames needs no account and no sign-up; what it records while your child plays is set out on the privacy page.