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AQASpec G10Higher tier~30 min

Circle Theorems

Mathematics · Topic revision workspace

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Read the notes, score 80% and know 75% of the cards.

Topic overview

What you need to know

Circle Theorems is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. The angle at the centre of a circle is twice the angle at the circumference when subtended by the same arc. The angle inscribed in a semicircle (with diameter as one side) is always 90°. Angles at the circumference subtended by the same arc are equal.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

  • Angle at Centre Theorem: The angle at the centre of a circle is twice the angle at the circumference when subtended by the same arc.
  • Angle in a Semicircle: The angle inscribed in a semicircle (with diameter as one side) is always 90°.
  • Angles in the Same Segment: Angles at the circumference subtended by the same arc are equal.
  • Cyclic Quadrilateral: A quadrilateral inscribed in a circle. Opposite angles sum to 180°.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

  • Tangent-Radius Relationship: A tangent to a circle is perpendicular to the radius at the point of contact (they meet at 90°).
  • Tangents from an External Point: Two tangents drawn from the same external point to a circle are equal in length.
  • Alternate Segment Theorem: The angle between a tangent and a chord equals the angle in the alternate segment.
  • Perpendicular from Centre to Chord: A perpendicular line from the centre of a circle to a chord bisects (halves) the chord.

Core knowledge

Key facts for Circle Theorems

1

Angle at Centre Theorem

The angle at the centre of a circle is twice the angle at the circumference when subtended by the same arc.

2

Angle in a Semicircle

The angle inscribed in a semicircle (with diameter as one side) is always 90°.

3

Angles in the Same Segment

Angles at the circumference subtended by the same arc are equal.

4

Cyclic Quadrilateral

A quadrilateral inscribed in a circle. Opposite angles sum to 180°.

5

Tangent-Radius Relationship

A tangent to a circle is perpendicular to the radius at the point of contact (they meet at 90°).

6

Tangents from an External Point

Two tangents drawn from the same external point to a circle are equal in length.

7

Alternate Segment Theorem

The angle between a tangent and a chord equals the angle in the alternate segment.

8

Perpendicular from Centre to Chord

A perpendicular line from the centre of a circle to a chord bisects (halves) the chord.

Active recall

Close the notes and answer these

  1. 1.Without looking, explain angle at centre theorem and give one example or consequence.
  2. 2.Without looking, explain angle in a semicircle and give one example or consequence.
  3. 3.Without looking, explain angles in the same segment and give one example or consequence.
  4. 4.Without looking, explain cyclic quadrilateral and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

Finished learning?

Mark the notes complete, then test what you can recall.

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