Circle Theorems
Master circle theorems for GCSE Maths with structured, exam-style practice. This Higher resource covers applying the circle theorems, with reasons and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Every answer needs a reason — a value alone won't score full marks.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
Circle theorems describe angle relationships involving circles. There are several, and questions usually require you to name the one you have used as well as apply it.
The most frequently used are: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is \(90^\circ\); angles in the same segment are equal; and opposite angles in a cyclic quadrilateral add to \(180^\circ\).
Two more involve tangents: a tangent meets a radius at \(90^\circ\), and two tangents from the same external point are equal in length. Learning the standard wording matters, because the reasoning mark depends on quoting the theorem correctly.
Revision notes
The main theorems
The angle at the centre is twice the angle at the circumference standing on the same arc. The angle in a semicircle is \(90^\circ\).
Angles in the same segment are equal, and opposite angles in a cyclic quadrilateral add to \(180^\circ\).
Tangent theorems
A tangent meets the radius at the point of contact at \(90^\circ\).
Two tangents drawn from the same external point are equal in length, which creates an isosceles triangle and often unlocks the rest of the question.
Writing the reasons
Quote the theorem in full, such as the angle at the centre is twice the angle at the circumference.
Abbreviations or vague phrases like circle theorem do not earn the reasoning mark.
Key points
- The angle at the centre is twice the angle at the circumference.
- The angle in a semicircle is \(90^\circ\).
- Angles in the same segment are equal.
- Opposite angles in a cyclic quadrilateral add to \(180^\circ\).
- A tangent meets a radius at \(90^\circ\).
- Two tangents from a point are equal in length.
Worked examples
Example 1
The angle at the circumference is \(35^\circ\). Find the angle at the centre on the same arc.
Working
Example 2
One angle of a cyclic quadrilateral is \(115^\circ\). Find the opposite angle.
Working
Example 3
A triangle is drawn in a semicircle with the diameter as its base. Find the angle at the circumference.
Working
Common mistakes
Not naming the theorem used.
The reasoning mark requires the theorem quoted in full, not just the arithmetic.
Halving instead of doubling at the centre.
The centre angle is twice the circumference angle, so double when going inwards.
Applying the same-segment rule across different arcs.
The angles must stand on the same arc for the rule to hold.
Assuming a quadrilateral is cyclic.
All four vertices must lie on the circle for the opposite-angle rule to apply.
Exam tips
- Learn the standard wording of each theorem.
- Mark angles on the diagram as you find them.
- Look for radii, tangents and diameters as clues to which theorem applies.
- Give the theorem as your reason every time.
Key terms
- Cyclic quadrilateral
- A quadrilateral with all four vertices on a circle.
- Segment
- The region between a chord and an arc.
- Tangent
- A line touching the circle at one point.
- Circumference
- The edge of the circle.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.