IGCSE Mathematics: Circle Theorems Practice Questions
The angle at the centre is twice the angle at the circumference on the same arc, the angle in a semicircle is 90 degrees, opposite angles of a cyclic quadrilateral add to 180 degrees, and a tangent meets a radius at 90 degrees.
Topic 4.7 of Cambridge IGCSE Mathematics 0580 awards marks for naming the theorem as well as for the number. An answer with no reason attached rarely scores full marks. The questions below make you state the reason every time.
What you need to know for Circle Theorems
- Angle at the centreThe angle subtended at the centre is twice the angle subtended at the circumference by the same arc.
- Angle in a semicircleAn angle at the circumference standing on a diameter is always 90 degrees.
- Angles in the same segmentAngles at the circumference subtended by the same arc, on the same side of it, are equal.
- Cyclic quadrilateralA quadrilateral with all four vertices on the circle. Opposite angles add to 180 degrees.
- Tangent and radiusA tangent meets the radius at the point of contact at 90 degrees. Two tangents drawn from the same external point are equal in length.
- Alternate segment theoremThe angle between a tangent and a chord equals the angle in the alternate segment.
IGCSE Mathematics Circle Theorems questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Points A and B lie on a circle with centre O. Angle ACB at the circumference is 35 degrees, where C is another point on the circle. Find angle AOB, giving a reason.
Show the worked answer
- Angle ACB is at the circumference and angle AOB is at the centre, both standing on the same arc AB.
- The angle at the centre is twice the angle at the circumference on the same arc.
- Angle AOB = 2 x 35 = 70 degrees.
- The reason must be stated in words for the reasoning mark.
AB is a diameter of a circle and C is a point on the circumference. Angle CAB is 28 degrees. Find angle ABC, giving reasons.
Show the worked answer
- Angle ACB stands on the diameter AB, so angle ACB = 90 degrees. The reason is that the angle in a semicircle is 90 degrees.
- The three angles of triangle ABC add to 180 degrees.
- Angle ABC = 180 minus 90 minus 28.
- Angle ABC = 62 degrees.
PQRS is a cyclic quadrilateral. Angle PQR is 105 degrees and angle QRS is 84 degrees. Find angles PSR and SPQ, giving a reason.
Show the worked answer
- Opposite angles of a cyclic quadrilateral add to 180 degrees.
- Angles PQR and PSR are opposite, so angle PSR = 180 minus 105 = 75 degrees.
- Angles QRS and SPQ are opposite, so angle SPQ = 180 minus 84 = 96 degrees.
- Check: the four angles total 105 + 84 + 75 + 96 = 360 degrees, as they must for any quadrilateral.
A tangent touches a circle at point T. A chord TB is drawn, and the angle between the tangent and the chord is 40 degrees. State the size of the angle in the alternate segment and name the theorem used.
Show the worked answer
- The angle between a tangent and a chord drawn from the point of contact is 40 degrees.
- The alternate segment is the region of the circle on the other side of the chord from that angle.
- The alternate segment theorem states that the angle between the tangent and the chord equals the angle subtended by the chord in the alternate segment.
- The angle in the alternate segment is therefore also 40 degrees.
Common mistakes in this topic
- Giving a numerical answer with no reason stated.
- Halving instead of doubling for the angle at the centre.
- Pairing adjacent angles in a cyclic quadrilateral.
- Assuming a triangle inside a circle is isosceles when the sides are chords rather than radii.
- Confusing the tangent and radius rule with the alternate segment theorem.
Exam tips
- Mark every radius on the diagram. Two radii always create an isosceles triangle, which often supplies the missing step.
- Write the reason next to every angle you find, in words. Reasoning marks are separate from answer marks.
- Look for a diameter first. If one is present, the angle in a semicircle gives you a free 90 degrees.
- Use the exact wording from the theorem list. Paraphrases sometimes fail to score.
- Check that a quadrilateral's four angles total 360 degrees before moving on.
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Circle Theorems FAQs
What are the main IGCSE circle theorems?
The angle at the centre is twice the angle at the circumference 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 to 180 degrees. A tangent meets a radius at 90 degrees, and the alternate segment theorem links a tangent and a chord.
Do I have to give reasons in circle theorem questions?
Yes. Cambridge awards separate marks for the reason, so a correct number without a stated theorem typically scores only part of the available marks. Write the reason in words alongside each angle you calculate, using the standard wording of the theorem.
How do I recognise a cyclic quadrilateral?
All four vertices must lie on the circumference of the circle. When they do, opposite angles add to 180 degrees. Angles that are next to each other have no such relationship, so identifying which pairs are opposite is the first step.
What is the alternate segment theorem?
The angle between a tangent and a chord drawn from the point of contact equals the angle subtended by that chord in the alternate segment, meaning the part of the circle on the other side of the chord. It is the theorem most often forgotten in exams.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.