Circle Theorems – Every Rule with Worked Examples

The Topic That Separates Good from Great

Circle theorems are one of the most commonly tested topics on the IGCSE 0580 Extended paper. They appear almost every year, often as a multi-part question worth six to eight marks. Many students find them challenging because there are several theorems to remember and exam questions often require you to combine multiple theorems in a single problem.

This guide covers every circle theorem you need for IGCSE Maths, with worked examples that mirror the style of real exam questions.

Theorem 1: Angle at the Centre Is Twice the Angle at the Circumference

When an arc of a circle subtends an angle at the centre and an angle at the circumference, the angle at the centre is exactly twice the angle at the circumference.

In practice: If you see a triangle formed by two radii and a chord, and another triangle formed by the same chord and a point on the circumference, the angle at the centre is double the angle at the circumference.

Worked Example

In a circle with centre O, points A, B, and C lie on the circumference. Angle AOB = 104°. Find angle ACB.

Solution:

Angle ACB = angle AOB ÷ 2 (angle at centre is twice angle at circumference)

Angle ACB = 104° ÷ 2 = 52°

Theorem 2: Angle in a Semicircle Is 90°

Any angle inscribed in a semicircle (that is, an angle subtended by a diameter at the circumference) is a right angle.

In practice: If you see a triangle inside a circle where one side is a diameter, the angle opposite the diameter is always 90°.

Worked Example

In a circle with centre O, AB is a diameter. C is a point on the circumference. Angle BAC = 35°. Find angle ABC.

Solution:

Angle ACB = 90° (angle in a semicircle)

Angle ABC = 180° - 90° - 35° = 55° (angles in a triangle)

Theorem 3: Angles in the Same Segment Are Equal

Angles subtended by the same arc at the circumference are equal. If two angles are both inscribed in the same segment of a circle (on the same side of a chord), they are equal.

In practice: If you see two triangles that share the same base chord and both have their apex on the same arc, the angles at the apex are equal.

Worked Example

Points A, B, C, and D lie on a circle. Angle ADB = 40°. Find angle ACB.

Solution:

Angle ACB = 40° (angles in the same segment)

Both angles are subtended by the same arc AB, so they must be equal.

Theorem 4: Opposite Angles of a Cyclic Quadrilateral Sum to 180°

A cyclic quadrilateral is a four-sided shape where all four vertices lie on a circle. The opposite angles of a cyclic quadrilateral add up to 180°.

Worked Example

ABCD is a cyclic quadrilateral. Angle A = 72° and angle B = 115°. Find angles C and D.

Solution:

Angle C = 180° - angle A = 180° - 72° = 108° (opposite angles of a cyclic quadrilateral)

Angle D = 180° - angle B = 180° - 115° = 65° (opposite angles of a cyclic quadrilateral)

Check: 72° + 115° + 108° + 65° = 360°. Correct.

Theorem 5: The Tangent to a Circle Is Perpendicular to the Radius

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

In practice: Whenever you see a tangent line touching a circle, draw the radius to that point and mark the 90° angle. This is often the key to unlocking the rest of the question.

Worked Example

A tangent to a circle at point A meets a line from the centre O at point B, where B is outside the circle. OA = 5 cm and OB = 13 cm. Find the length of the tangent AB.

Solution:

Angle OAB = 90° (tangent is perpendicular to radius)

Using Pythagoras’ theorem in triangle OAB:

AB² = OB² - OA²

AB² = 169 - 25 = 144

AB = 12 cm

Theorem 6: Two Tangents from an External Point Are Equal

If two tangent lines are drawn to a circle from the same external point, their lengths (from the external point to the points of tangency) are equal.

Worked Example

From a point P outside a circle with centre O, two tangents are drawn to the circle touching it at A and B. PA = 8 cm and OA = 6 cm. Find angle APB.

Solution:

First, find OP using Pythagoras (angle OAP = 90° since tangent is perpendicular to radius):

OP² = OA² + PA² = 36 + 64 = 100

OP = 10 cm

Since PA = PB (tangents from external point are equal), triangle OAP is congruent to triangle OBP.

Find angle OPA:

tan(OPA) = OA/PA = 6/8 = 0.75

Angle OPA = tan⁻¹(0.75) = 36.87°

Angle APB = 2 × 36.87° = 73.7° (to 1 decimal place)

Theorem 7: The Alternate Segment Theorem

The angle between a tangent to a circle and a chord drawn from the point of tangency is equal to the angle in the alternate segment.

This is the theorem that students find most confusing, but it is simply about recognising the pattern. The angle between the tangent and the chord equals the angle subtended by the chord at the circumference on the opposite side.

Worked Example

A tangent at point A on a circle meets a chord AB at point A. The angle between the tangent and chord AB is 55°. C is a point on the major arc. Find angle ACB.

Solution:

Angle ACB = 55° (alternate segment theorem)

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

Putting It All Together: Multi-Step Example

Points A, B, C, and D lie on a circle with centre O. A tangent to the circle at A meets line BC extended at point T. Angle BAT = 50° and angle ADC = 70°.

Find angle ABC.

Step 1: Angle ABC + angle ADC = 180° (opposite angles of cyclic quadrilateral ABCD)

Angle ABC = 180° - 70° = 110°

Find angle ACB.

Step 2: Angle ACB = angle BAT = 50° (alternate segment theorem)

Find angle BAC.

Step 3: In triangle ABC:

Angle BAC = 180° - 110° - 50° = 20° (angles in a triangle)

Exam Tips for Circle Theorems

1. Always state the theorem you are using. The mark scheme awards marks for identifying the correct theorem by name. Write it in brackets after your calculation.
2. Look for the centre. If the centre O is marked, look for radii. Remember that all radii are equal, which creates isosceles triangles.
3. Look for tangent lines. Mark the 90° angle between the tangent and radius immediately.
4. Look for cyclic quadrilaterals. If four points are on the circle, check whether opposite angles sum to 180°.
5. Draw on the diagram. Add any lines that help you see the relationships — radii, diameters, and connections to the centre.
6. Work systematically. In multi-part questions, each part usually provides information you need for the next part.

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