Circle TheoremsCambridge IGCSE Maths: Revision notes

Section 1

What circle vocabulary do I need?

Before using circle theorems, be confident with circle vocabulary:

Centre: the fixed point equidistant from every point on the circle
Radius: a line from the centre to the circumference
Diameter: a chord passing through the centre (twice the radius)
Chord: a straight line joining two points on the circumference
Tangent: a straight line that touches the circle at exactly one point
Arc: part of the circumference; sector: a 'pie-slice' region between two radii and an arc; segment: a region between a chord and an arc

Key termschordtangentsegment

Section 2

What are the two basic circle theorems (Core and Extended)?

1. Angle in a semicircle = 90° Any angle drawn from the two ends of a diameter to a point on the circumference is a right angle.

2. Angle between a tangent and a radius = 90° Wherever a tangent touches a circle, the radius drawn to that point meets the tangent at exactly 90°.

These two facts alone can unlock many exam questions when combined with angle sum rules for triangles and quadrilaterals.

Key termsangle in a semicircletangent-radius angle

Example

If AB is a diameter and C is any other point on the circle, angle ACB = 90° every time, regardless of where C sits on the circumference.

Section 3

What further circle theorems apply at Extended level?

Theorem	Statement
Angle at centre	The angle at the centre is twice the angle at the circumference, standing on the same arc
Angles in the same segment	Angles subtended by the same arc, in the same segment, are equal
Cyclic quadrilateral	Opposite angles of a cyclic quadrilateral (a quadrilateral with all four vertices on the circle) sum to 180°
Alternate segment theorem	The angle between a tangent and a chord equals the angle in the alternate segment

Each theorem must be stated correctly by name when giving a geometric reason.

Key termscyclic quadrilateralalternate segment theorem

Exam tip

Examiners expect the theorem name in your reasoning, e.g. 'angle at centre = 2 × angle at circumference', not just the numerical answer.

Section 4

What symmetry properties of circles are useful (Extended)?

Equal chords are equidistant from the centre — if two chords are the same length, they are the same perpendicular distance from the centre
The perpendicular bisector of a chord passes through the centre — useful for locating a circle's centre or finding chord midpoints
Tangents drawn from an external point are equal in length — if two tangents touch a circle from the same outside point, both tangent segments are the same length, and the line joining the external point to the centre bisects the angle between the tangents

Key termsperpendicular bisector

Common mistake

Forgetting that tangents from one external point are equal often costs marks in isosceles-triangle-style tangent questions.

Section 5

How do I structure a circle theorem answer?

State the theorem you are using by name
Show the calculation clearly, one line at a time
Give the final angle with its unit (degrees)

Worked example: A cyclic quadrilateral has opposite angles $x$ and 108°. Find $x$ . Opposite angles of a cyclic quadrilateral sum to 180°, so $x = 180 - 108 = 72$ °.

Example

Angle at the centre is 140°. The angle at the circumference on the same arc is 140° ÷ 2 = 70°.

Must Know

Angle in a semicircle = 90°
Angle between tangent and radius = 90°
Angle at centre = 2 × angle at circumference (same arc)
Angles in the same segment are equal
Opposite angles of a cyclic quadrilateral sum to 180°
Alternate segment theorem: tangent–chord angle = angle in alternate segment
Tangents from the same external point are equal in length

That's the notes covered.

Carry on to the next subtopic.