Higher – Calculating angles using circles

Part ofMathsGeometry and measure

Key points about calculating angles using circles

Bullet points represented by lightbulbs

  • Circles have different angle properties, described by a number of theorems. This is one of two guides on circle theorems, with this guide covering the following:

    • the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference.

    • the angle subtended at the circumference by a semicircle is a right angle.

    • the angle at the circumference of a semi-circle that is subtended by the diameter is a right angle.

    • opposite angles of a cyclic quadrilateral add up to 180°.

When you are confident with the theorems in this guide, have a look at the guide on How to calculate tangents and chords using the alternate segment theorem.

Make sure you are confident using other angles rules, such as angles on a straight line add up to 180° and that angles in a triangle add up to 180°, and can recognise angles in parallel lines.

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Angle subtended by an arc

An angle created by lines drawn from the endpoints of an arc to a specific point, is said to be subtended by an arc

When an angle is created by lines drawn from the endpoints of an to a specific point it is said to be by an arc.

The angle at the centre of a circle is twice the angle at the that is subtended by the same arc.

Angle AOB in the image is twice the size of angle ACB.

Line segments AO and OB are as they are the same length. A consequence of this is there can be embedded within the diagram. Triangle AOB is isosceles.

Find out more, along with worked examples

GCSE exam-style questions

A pen and a piece of paper with question marks on it.
  1. Calculate the size of angle 𝑗.
An angle of 57 degrees is subtended from angle j at the centre of the circle

  1. Calculate the size of angle 𝑘.
The angle at the centre of the circle is 86 degrees. The angle at the circumference creates a straight line with angle k

  1. Calculate the size of angle 𝑙.
Angle l (ACB) is subtended from the centre of the circle. A triangle is formed between the centre of the circle and points A and B. One of the angles of the triangle at the circumference is 32 degrees

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Angle subtended at the circumference by a semicircle

A triangle formed along the diameter of a circle forms a 90 degree angle at the circumference

  • The angle at the of a that is subtended by the is a right angle.

  • For any circle with diameter AB, if point C lies on the circumference of the circle, then angle ACB = 90°.

A triangle formed along the diameter of a circle forms a 90 degree angle at the circumference

Follow the worked example

GCSE exam-style questions

  1. By forming and solving an equation, work out the value of 𝑥.
Triangle formed from the diameter of a circle, with one unknown angle touching the circumference and angles of 2x and 3x

  1. Is line segment MN a diameter?
Triangle within a circle with angles of 48 degrees, 46 degrees and unknown angle

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Angles in the same segment

Angles subtending from each point on each end of the same segment are equal

Angles at the in the same are equal.

The vertices C and D are on the circumference and are subtended by the same arc AB.

Angle ACB = Angle ADB

Angles subtending from each point on each end of the same segment are equal

Find out more, along with a worked example

GCSE exam-style questions

A pen and a piece of paper with question marks on it.
  1. Calculate the size of angle 𝑦.
A circle with four points labelled E, F, G, and H on its circumference. Two intersecting chords, GE and FH, form triangles inside the circle. The angle at point G between chords GE and GF is marked 32°, the angle at point E between chords GE and EH is marked 51°, and the angle at point H between chords HF and HE is labelled y.

  1. Calculate the size of angle 𝑤.
A circle with centre labelled O and four points on its circumference labelled A, B, C, and D. Point C is at the top, A at the bottom, B on the right, and D on the left. A vertical diameter runs from A to C through O. Chords connect A to B, A to D, and B to C, forming triangles inside the circle. The angle at point D between chords DA and DC is marked 27°, and the angle at point A between chords AB and AD is labelled w.

  1. In the diagram, WZ is parallel to XY. Calculate the size of angle 𝑎.
A circle with four points on its circumference labelled W, X, Y, and Z. Point W is at the bottom, Z at the top left, Y at the top right, and X at the right side. Two intersecting chords form triangles inside the circle: WZ and XY. The angle at point Z between chords ZW and ZX is marked 52°, and the angle at point W between chords WZ and WX is labelled a. Both chords WZ and XY have arrow markings indicating they are parallel.

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Opposite angles in a cyclic quadrilateral

A cyclic quadrilateral, where vertices A, B, C and D all touch the circumference

A where all four lie on the circumference of a circle is called a

The opposite angles in a cyclic quadrilateral add up to 180°.

Angle A + Angle C = 180°

Angle B + Angle D = 180°

A cyclic quadrilateral, where vertices A, B, C and D all touch the circumference

Follow the worked example

GCSE exam-style questions

A pen and a piece of paper with question marks on it.
  1. Calculate the size of angle 𝑓.
A circle with points W, X, Y, and Z on its circumference. Point V is outside the circle. Lines connect V to X and Y, forming an angle of 41° at V. Inside the circle, lines connect W to X and Z, forming a triangle WXZ. Point Y lies on the arc between X and Z, and angle f is marked at Y between lines XY and YZ. Angle at Z inside the circle is labeled 95°

  1. Calculate the size of angle 𝑥.
A circle with points A, B, C, and D on its circumference and point O at the centre. Lines connect A, B, C, and D forming a quadrilateral inside the circle. Angle at A is labelled 126° and marked with x. Point O connects to A and B with two radii forming an angle at the centre.

  1. The diagram shows a cyclic quadrilateral.

𝑥 : 𝑦 = 2 : 1

Calculate the size of angle 𝑧.

Angles in a cyclic quadrilateral equal x, y, z and x minus 20 degrees

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Check your understanding

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Quiz – Calculating angles with circles

Practise what you've learned about calculating angles with circles with this quiz.

Now you've revised calculating angles with circles, why not look at circumference and arc length?

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