Circle Theorems

Before applying the theorems, the correct terminology must be secure. All eight circle theorems describe relationships between these parts.

Term	Definition
Centre	The fixed point equidistant from every point on the circle
Radius	A straight line from the centre to any point on the circumference
Diameter	A chord passing through the centre; equal to 2 × radius
Chord	A straight line joining two points on the circumference (not through the centre)
Arc	Part of the circumference between two points
Tangent	A straight line that touches the circumference at exactly one point
Sector	The region bounded by two radii and the arc between them
Segment	The region between a chord and the arc it cuts off

Circle theorems (G10) are Higher tier only. Foundation students need only the definitions above (G9).

A cyclic polygon has all its vertices on the circumference. A cyclic quadrilateral is a four-sided cyclic polygon — its four vertices all lie on the circle.

Theorem 1 — Angle at the centre: The angle subtended at the centre by an arc is twice the angle subtended at the circumference by the same arc.

$\angle \text{centre}=2\times \angle \text{circumference (same arc)}$

Worked example: If the angle at the centre is 80°, the angle at the circumference on the same arc is 40°. If the angle at the circumference is 35°, the angle at the centre is 70°.

Theorem 2 — Angle in a semicircle: An angle at the circumference subtended by a diameter is always 90°.

This is a special case of Theorem 1: the angle at the centre is 180° (a straight line = the diameter), so the angle at the circumference is $180°÷2=90°$.

If you see…	Apply…
An angle at the centre and an angle at the circumference on the same arc	Theorem 1: centre = 2 × circumference
An angle in a semicircle (one side is the diameter)	Theorem 2: angle = 90°

When using Theorem 1, confirm that both angles are subtended by the same arc. The angle at the circumference on the opposite arc uses the reflex angle at the centre, not the same one.

Theorem 3 — Angles in the same segment: Angles subtended by the same arc at the circumference are equal.

If two angles both stand on the same chord from the same side, they are equal — regardless of where exactly on the arc the vertex is placed.

Worked example: Points A, B, C, D lie on a circle. Angle ADB = 42°. Find angle ACB if C is on the same arc as D.

Since angles ADB and ACB are in the same segment (both subtended by chord AB from the same arc): $\angle ACB=\angle ADB=42°$.

Theorem 4 — Cyclic quadrilateral: Opposite angles in a cyclic quadrilateral sum to 180°.

$\angle A+\angle C=180°\angle B+\angle D=180°$

Worked example: In cyclic quadrilateral ABCD, angle A = 73° and angle B = 105°. Find angles C and D.

$\angle C=180°-73°=107°\angle D=180°-105°=75°$

Check: $73°+107°=180°$ ✓ and $105°+75°=180°$ ✓

Theorem 5 — Tangent-radius angle: A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

$\text{tangent}\perp \text{radius at the point of contact}⟹\angle =90°$

Theorem 6 — Two tangents from an external point: Two tangent lines drawn from the same external point are equal in length.

If PT and PQ are both tangents from external point P to the same circle, then $PT=PQ$.

Worked example: Two tangents from point P touch a circle at points A and B. The angle APB = 50°. Find the angle AOB where O is the centre.

Since PA and PB are tangents: $\angle OAP=\angle OBP=90°$ (Theorem 5).

In quadrilateral OAPB: angles sum to 360°.

$\angle AOB=360°-90°-90°-50°=130°$

Theorem 5 creates a right angle that is not always drawn in exam diagrams. Mark it in yourself before writing any equations.

Theorem 7 — Perpendicular from centre to a chord: The perpendicular from the centre of a circle to a chord bisects the chord.

Equivalently: the line from the centre to the midpoint of a chord is perpendicular to the chord.

Worked example: A chord of length 12 cm is 5 cm from the centre of a circle. Find the radius.

The perpendicular from the centre bisects the chord, creating a right-angled triangle with legs 5 cm and 6 cm (half of 12 cm).

$r=\sqrt{5^2+6^2}=\sqrt{25+36}=\sqrt{61}\approx 7.81\text{ cm}$

Theorem 8 — Alternate segment theorem: The angle between a tangent and a chord drawn from the point of tangency equals the angle subtended by that chord in the alternate segment.

In other words: the angle in the triangle formed by the tangent and chord equals the angle in the opposite part of the circle.

Worked example: A tangent meets a circle at point T. A chord TB makes an angle of 48° with the tangent. Find the angle subtended by TB in the alternate segment.

By the alternate segment theorem: the angle in the alternate segment = 48°.

1. Confusing angle at the centre with angle at the circumference

The angle at the centre is twice the angle at the circumference — students frequently invert this and halve the circumference angle or double the centre angle in the wrong direction. Draw both angles clearly and label which is which before writing the equation.

2. Forgetting to check that both angles share the same arc

Theorem 1 and Theorem 3 only apply when the angles are subtended by the same arc. An angle on the major arc and an angle on the minor arc use different versions of the relationship.

3. Not using Theorem 5 to create a right angle

Tangent questions almost always require marking the right angle between the tangent and the radius. Students who do not mark it fail to see the right-angled triangle they need for Pythagoras or trigonometry.

4. Misidentifying the alternate segment

The alternate segment is on the opposite side of the chord from the angle you are reading. Drawing a clear diagram and shading the alternate segment before applying Theorem 8 prevents errors.

5. Incomplete reasoning in proof questions

When asked to prove a circle theorem result, each step must reference a theorem by name. "Angle in the same segment" is not enough — write "Angles in the same segment are equal". Missing reasons lose the reasoning marks even when the algebra is correct.