Circle Theorems [H]

This lesson covers the circle theorems required for the Higher tier of AQA GCSE Mathematics. Circle theorem questions are popular on exam papers because they test your ability to apply multiple rules and justify your reasoning. You must learn each theorem, be able to recognise when to use it, and give the correct mathematical reason with your answer.

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Key Circle Vocabulary

Term	Definition
Radius	A straight line from the centre to the circumference
Diameter	A straight line through the centre, from one side of the circle to the other (= 2 x radius)
Chord	A straight line connecting two points on the circumference
Tangent	A straight line that touches the circle at exactly one point
Arc	A part of the circumference
Sector	The region between two radii and an arc
Segment	The region between a chord and an arc
Subtend	An angle is "subtended" by an arc at a point — it means the angle is formed at that point by lines drawn to the ends of the arc

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The Circle Theorems

Theorem 1: Angle at the Centre

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

graph TD
    A[Same arc] --> B[Angle at centre = 2 x angle at circumference]

Worked Example 1

The angle at the centre is 140 degrees. Find the angle at the circumference subtended by the same arc.

Solution: Angle at circumference = 140 / 2 = 70 degrees Reason: The angle at the centre is twice the angle at the circumference.

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Theorem 2: Angle in a Semicircle

The angle in a semicircle is always 90 degrees.

(This is a special case of Theorem 1 where the angle at the centre is 180 degrees, so the angle at the circumference is 90 degrees.)

Worked Example 2

Triangle ABC is inscribed in a circle where AB is a diameter. Angle BAC = 37 degrees. Find angle ABC.

Solution: Angle ACB = 90 degrees (angle in a semicircle) Angle ABC = 180 - 90 - 37 = 53 degrees (angles in a triangle sum to 180 degrees)

Exam Tip: Whenever you see a triangle inscribed in a circle with one side being the diameter, immediately mark the angle opposite the diameter as 90 degrees. This is the most commonly used circle theorem.

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Theorem 3: Angles in the Same Segment

Angles in the same segment (subtended by the same arc, on the same side of the chord) are equal.

Worked Example 3

Two angles are subtended by the same chord on the same side. One is 48 degrees. Find the other.

Solution: The other angle = 48 degrees Reason: Angles in the same segment are equal.

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Theorem 4: Opposite Angles in a Cyclic Quadrilateral

The opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle with all four vertices on the circumference) sum to 180 degrees.

Worked Example 4

In a cyclic quadrilateral ABCD, angle A = 110 degrees and angle B = 75 degrees. Find angles C and D.

Solution: Angle C = 180 - 110 = 70 degrees (opposite angles in a cyclic quadrilateral sum to 180 degrees) Angle D = 180 - 75 = 105 degrees (opposite angles in a cyclic quadrilateral sum to 180 degrees)

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Theorem 5: Tangent is Perpendicular to the Radius

A tangent to a circle is perpendicular to the radius at the point of contact.

This means the angle between the tangent and the radius is always 90 degrees.

Worked Example 5

A tangent meets a circle at point P. The radius OP makes an angle of 90 degrees with the tangent. A line from P to another point on the circle, Q, makes an angle of 55 degrees with the tangent. Find angle OPQ.

Solution: Angle OPQ = 90 - 55 = 35 degrees

Exam Tip: When you see a tangent in a diagram, immediately look for the radius to the point of tangency and mark the right angle. This is often the key to unlocking the rest of the problem.

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Theorem 6: Two Tangents from an External Point

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

The line from the external point to the centre bisects the angle between the tangents.

Worked Example 6

Two tangents are drawn from point T to a circle with centre O, touching at A and B. TA = 8 cm. Find TB.

Solution: TB = 8 cm (two tangents from an external point are equal in length)

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Theorem 7: Perpendicular from Centre to Chord

The perpendicular from the centre of a circle to a chord bisects the chord.

Conversely, a line from the centre to the midpoint of a chord is perpendicular to the chord.

Worked Example 7

A chord of length 24 cm is 5 cm from the centre of a circle. Find the radius.

Solution: The perpendicular from the centre bisects the chord, so each half = 12 cm. Using Pythagoras: r squared = 12 squared + 5 squared = 144 + 25 = 169 r = square root of 169 = 13 cm

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Theorem 8: 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.

This is often the trickiest theorem to spot. The angle between the tangent and chord equals the angle subtended by the chord in the opposite segment.

Worked Example 8

A tangent at point P makes an angle of 62 degrees with chord PQ. Find the angle PRQ, where R is on the major arc.

Solution: Angle PRQ = 62 degrees (alternate segment theorem)

Exam Tip: The alternate segment theorem is the one students most commonly forget or fail to spot. Look for it whenever you see a tangent meeting a chord. The angle between them equals the angle "on the other side" of the chord.

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Proving Circle Theorems

You may be asked to prove circle theorems. The key tools are:

1. Isosceles triangles — any two radii form an isosceles triangle
2. Angles in a triangle sum to 180 degrees
3. Base angles of an isosceles triangle are equal

Proof Sketch: Angle at the Centre = 2 x Angle at Circumference

Let O be the centre and let angle at the circumference be subtended at point C. Draw radii OA and OB to the ends of the arc, and draw radius OC.

Triangles OAC and OBC are both isosceles (OA = OC = OB = radius).

Let angle OCA = angle OAC = x (base angles of isosceles triangle OAC) Let angle OCB = angle OBC = y (base angles of isosceles triangle OBC)

Angle at circumference = x + y Angle AOC = 180 - 2x (angles in triangle OAC) Angle BOC = 180 - 2y (angles in triangle OBC) Angle at centre AOB = 360 - (180 - 2x) - (180 - 2y) = 2x + 2y = 2(x + y)

Therefore, angle at centre = 2 x angle at circumference.

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Circle Theorems Summary Table

Theorem	Statement
Angle at the centre	= 2 x angle at circumference (same arc)
Angle in a semicircle	= 90 degrees
Same segment	Angles subtended by the same arc in the same segment are equal
Cyclic quadrilateral	Opposite angles sum to 180 degrees
Tangent-radius	Tangent is perpendicular to radius (90 degrees)
Two tangents	Equal length from an external point
Perpendicular to chord	From centre, bisects the chord
Alternate segment	Angle between tangent and chord = angle in alternate segment

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Summary