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Circle theorems are properties that are true for all circles, regardless of their size.

There are six theorems to learn and recognise. Questions can have a combination of theorems.

It is important to know the theorems well and to learn to identify them in the diagrams.

An important word that is used in circle theorems is subtend.

An angle is created by two lines.

The angle in between the two lines is subtended by the arc between C and D.

The angle subtended by an arc at the centre is twice the angle subtended at the circumference.

More simply, the angle at the centre is double the angle at the circumference.

Example

Calculate the angles x and y.

Solution

Using theorem 1

The angle at the centre is double the angle at the circumference.

• x = 2 x 50 = 100°
• y = 2 x 40 = 80°

The angle at the circumference in a semicircle is a right angle.

Angle APB = 90°

This is because the angle at the centre is 180° and the angle on the circumference is half the angle in the centre, 90°.

Example

Calculate the angle z.

Solution:

Using theorem 2

The angle at the circumference in a semicircle is a right angle.

• Angle STU = 90°
• z = 180 – 90 – 31 = 59°

Answer:

z = 59°

The angles at the circumference subtended by the same arc are equal.

More simply, angles in the same segment are equal.

Example

What is the size of the angles p and q?

Solution:

Using theorem 3

Angles in the same segment are equal.

• p = 52°
• q = 40°

Example

Calculate the size of angle b.

Using theorem 3.

Angles in the same segment are equal.

Using theorem 2.

The angle at the circumference in a semicircle is a right angle.

• b = 90 – 54
• b = 36°

A cyclic quadrilateral is a quadrilateral inside a circle with all 4 vertices touching the circumference of the circle.

The opposite angles are supplementary.

More simply, opposite angles in a cyclic quadrilateral add up to 180°.

• a + c = 180°
• b + d = 180°

Example

Calculate angles a and b.

Solution:

All 4 vertices touch the circumference of the circle.

Using theorem 4.

Opposite angles in a cyclic quadrilateral add up to 180°

a = 180 – 60 = 120°
b = 180 – 140 = 40°

Tangents

A tangent is a straight line which touches the circle at a point but does not cut through the circle.

There are two theorems involving tangents.

The angle between a tangent and a radius is 90°.

Example

Calculate the size of the angle BOC.

Solution:

Using theorem 5

The angle between a tangent and a radius is 90°..

Angle OCA and OBA are both 90°.

OB = OC because they are both radii of the circle.

Therefore, ABOC is a kite..

Angle BOC = 360 – 90 – 90 – 50 = 130°

Answer:

BOC = 130°

A straight line that just touches a point on a curve is known as a tangent.

A straight line joining two points on a circle is known as a chord.

The angle between a tangent and a chord is equal to the angle in the alternate segment.

Example

Calculate the size of angle x.

Using theorem 6

The angle between a tangent and a chord is equal to the angle in the alternate segment.

Since angles in a triangle add to 180°

• x = 180 – 62 – 44
• x = 74°

Circle theorems can be used to solve more complex problems which require multiple theorems.