IGCSE Mathematics · 0580

IGCSE Mathematics: Circle Theorems Practice Questions

Circle theorem questions are quick marks once you can spot which rule applies. These worked examples cover the angle at the centre, the angle in a semicircle and opposite angles in a cyclic quadrilateral.

What you need to know

The angle at the centre is twice the angle at the circumference standing on the same arc.
The angle in a semicircle is 90 degrees.
Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral add up to 180 degrees.

The angle at the centre is twice the angle at the circumference on the same arc.

Practice questions with answers

Question 12 marks

The angle at the centre of a circle standing on an arc is 120 degrees. Find the angle at the circumference standing on the same arc.

The angle at the centre is twice the angle at the circumference. So the angle at the circumference = 120 divided by 2 = 60 degrees.

Mark schemeRecall angle at centre is twice angle at circumference (1). Answer 60 degrees (1).

Common mistakeDoubling instead of halving, giving 240 degrees.

Exam tipState the theorem you are using. Cambridge often awards a mark for the correct reason.

Question 22 marks

A triangle is drawn in a circle so that one side is a diameter. State the size of the angle opposite the diameter, and give the reason.

The angle is 90 degrees. The reason is that the angle in a semicircle is a right angle.

Mark schemeAngle is 90 degrees (1). Reason, angle in a semicircle (1).

Common mistakeNot giving the reason, which is a required mark in most circle theorem questions.

Exam tipAlways write the reason. Reasons are examined explicitly in this topic.

Question 32 marks

A cyclic quadrilateral has one angle of 95 degrees. Find the size of the opposite angle.

Opposite angles of a cyclic quadrilateral add up to 180 degrees. So the opposite angle = 180 minus 95 = 85 degrees.

Mark schemeRecall opposite angles sum to 180 (1). Answer 85 degrees (1).

Common mistakeAssuming opposite angles are equal rather than adding to 180.

Exam tipOpposite angles of a cyclic quadrilateral sum to 180, not to each other.

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Frequently asked questions

What is the angle at the centre theorem?

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

Why is the angle in a semicircle 90 degrees?

It is a special case of the angle at the centre theorem. The diameter gives a central angle of 180 degrees, so the angle at the circumference is 90 degrees.

What is a cyclic quadrilateral?

A cyclic quadrilateral is a four-sided shape with all four vertices on the circumference of a circle. Its opposite angles add to 180 degrees.

Do I need to give reasons in circle theorem questions?

Yes. Most circle theorem marks require you to state the theorem you used as well as the final answer.

More IGCSE Mathematics practice

Quadratic equations Pythagoras theorem