Circle Theorems
Circle theorems are a set of rules that describe the relationship between angles and lines within a circle. Mastering them is a key higher-tier skill, essential for solving complex geometry problems in architecture, engineering, and your GCSE Maths exam.
The Key Circle Theorems
1. Angle at the Centre
The angle at the centre is twice the angle at the circumference when both are subtended by the same arc.2. Angles in the Same Segment
Angles subtended by the same arc at the circumference are equal.3. Angle in a Semicircle
The angle in a semicircle is always a right angle ($90^\circ$).4. Cyclic Quadrilaterals
Opposite angles of a cyclic quadrilateral (all four vertices on the circumference) sum to $180^\circ$.5. Radius and Tangent
The angle between a tangent and a radius at the point of contact is $90^\circ$.6. Tangents from a Point
Two tangents drawn from an external point to a circle are of equal length.7. Alternate Segment Theorem
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment (the inscribed angle on the other side of the chord).Worked Example
Find angles x and y. O is the centre.
| Statement | Reason |
| $x = 130^\circ \div 2 = 65^\circ$ | The angle at the circumference is half the angle at the centre. |
| $\triangle OAC$ is an isosceles triangle. | OA and OC are both radii. |
| $\angle OAC = \angle OCA = y$ | Base angles of an isosceles triangle are equal. |
| $y = (180^\circ – 130^\circ) \div 2 = 25^\circ$ | Angles in a triangle sum to $180^\circ$. |
Tutor Insights
🤔 Common Misunderstandings
- Identifying the “Same Arc”: Struggling to see which angles at the circumference or centre are subtended by the same arc.
- Alternate Segment Theorem: This is often seen as the trickiest theorem. A good tip is to trace the angle between the tangent and the chord, then find the angle in the triangle that doesn’t touch the point of contact.
📝 Common Exam Mistakes
- Missing reasons. This is the biggest cause of lost marks. You MUST state the full name of the theorem you are using.
- Applying the wrong theorem. For example, using the cyclic quadrilateral rule for a shape that isn’t cyclic.
- Assuming a line is a diameter when it’s not stated or proven.
Practice Questions
- Find the value of angle $x$. AB is the diameter of the circle.
- Find the value of angle $y$. PQRS is a cyclic quadrilateral.
Show Answers
- $x = 90^\circ$. Reason: The angle in a semicircle is a right angle.
- $y = 95^\circ$. Reason: Opposite angles of a cyclic quadrilateral sum to $180^\circ$. ($180 – 85 = 95$).
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