WAEC / WASSCE
WAEC Circle Geometry Practice Questions, Fully Solved
WAEC circle geometry questions test a small set of theorems: the angle at the centre is twice the angle at the circumference, angles in the same segment are equal, opposite angles of a cyclic quadrilateral sum to 180 degrees, the angle in a semicircle is 90 degrees, and tangent properties. Here are five original questions in WASSCE style, solved with reasons stated the way examiners expect.
Circle geometry frightens people because the diagrams look busy, but the theory behind a WAEC circle question is a short list of theorems that never changes. Once you can name which theorem a diagram is using, the marks come quickly, and WAEC actually awards marks for stating the reason, not just the number. The five questions below are original, written in the WASSCE style rather than copied from past papers. I sit Cambridge exams myself, and circle theorems are one of the few topics that are almost identical across both systems, so this is familiar ground.
The circle theorems WAEC actually tests
Five theorems cover nearly every WASSCE circle question. First, the angle a chord subtends at the centre is twice the angle it subtends at the circumference on the same arc. Second, angles in the same segment, standing on the same chord, are equal. Third, opposite angles of a cyclic quadrilateral add up to 180 degrees. Fourth, the angle in a semicircle is a right angle. Fifth, tangent facts: a tangent is perpendicular to the radius at the point of contact, and the angle between a tangent and a chord equals the angle in the alternate segment. Add the isosceles triangle formed by two radii and you can unlock almost any diagram WAEC draws.
How to earn the marks, not just the answer
WAEC circle questions usually say give reasons for your answers, and the mark scheme splits marks between the value and the reason. So write angle ACB equals 64 degrees, angle at centre is twice angle at circumference, on one line. The exact wording is flexible but there must be a recognisable theorem, not just because of the circle. Second habit: diagrams in the paper are not drawn to scale, so never measure with a protractor; an answer that only a measurement could produce scores zero. Third, when a diagram has two radii to points on the circle, mark the isosceles triangle immediately. That one observation is the hidden step in a large share of these questions.
Worked questions, step by step
A, B and C are points on a circle with centre O, with C on the major arc AB. Angle AOB = 128 degrees. Find, with reasons, (a) angle ACB, (b) angle OAB.
- Part (a): the angle at the centre is twice the angle at the circumference standing on the same arc AB.
- So angle ACB equals 128 divided by 2 equals 64 degrees.
- Part (b): OA and OB are radii, so triangle OAB is isosceles with angle OAB equal to angle OBA.
- Angles in a triangle sum to 180: angle OAB equals (180 minus 128) divided by 2 equals 26 degrees.
Answer: (a) angle ACB = 64 degrees (b) angle OAB = 26 degrees
Where marks slip: Each part is typically value plus reason: A1 for 64 with B1 for naming the centre theorem, then M1 for the isosceles observation and A1 for 26. Reasons left out can cost half the marks.
Try one yourself: With the same setup, angle AOB = 146 degrees. Find angle ACB. [Answer: 73 degrees]
PQRS is a cyclic quadrilateral. Angle QPS = 74 degrees and angle PQR = 112 degrees. Find, with reasons, angle QRS and angle PSR.
- Opposite angles of a cyclic quadrilateral are supplementary, they add to 180 degrees.
- Angle QRS is opposite angle QPS: 180 minus 74 equals 106 degrees.
- Angle PSR is opposite angle PQR: 180 minus 112 equals 68 degrees.
- Check: 74 plus 112 plus 106 plus 68 equals 360, the angle sum of a quadrilateral. Correct.
Answer: angle QRS = 106 degrees, angle PSR = 68 degrees
Where marks slip: A1 plus reason for each angle. The quickest self-check is that all four angles must total 360 degrees; if they do not, one of your opposite pairs is wrong.
Try one yourself: In cyclic quadrilateral ABCD, angle A = 81 degrees and angle B = 95 degrees. Find angles C and D. [Answer: angle C = 99 degrees, angle D = 85 degrees]
TA is a tangent to a circle at A, and AB is a chord. C is a point on the major arc AB, and O is the centre. Angle TAB = 58 degrees. Find, with reasons, (a) angle ACB, (b) angle AOB.
- Part (a): the angle between a tangent and a chord equals the angle in the alternate segment.
- So angle ACB equals angle TAB equals 58 degrees.
- Part (b): the angle at the centre is twice the angle at the circumference on the same arc.
- So angle AOB equals 2 times 58 equals 116 degrees.
Answer: (a) angle ACB = 58 degrees (b) angle AOB = 116 degrees
Where marks slip: The alternate segment theorem is the least-known theorem on the list, and examiners reward naming it. A1 plus B1 for each part; quoting tangent perpendicular to radius here is the wrong reason and loses the reason mark.
Try one yourself: A tangent at P makes an angle of 47 degrees with chord PQ. Find the angle in the alternate segment. [Answer: 47 degrees]
AB is a diameter of a circle and C is a point on the circle. Angle CAB = 37 degrees. Find, with reasons, angle CBA.
- The angle in a semicircle is a right angle, so angle ACB equals 90 degrees.
- Angles in a triangle sum to 180 degrees.
- Angle CBA equals 180 minus 90 minus 37 equals 53 degrees.
Answer: angle CBA = 53 degrees
Where marks slip: B1 for stating angle ACB is 90 with the semicircle reason, then A1 for 53. The word diameter in a question is almost always a signal to use this theorem first.
Try one yourself: AB is a diameter and angle CAB = 29 degrees. Find angle CBA. [Answer: 61 degrees]
A chord of a circle is 16 cm long and its distance from the centre is 6 cm. Calculate (a) the radius of the circle, (b) the circumference, taking pi = 3.142, correct to one decimal place.
- The perpendicular from the centre bisects the chord, giving half-length 8 cm.
- The radius, half-chord and distance form a right triangle: r^2 equals 8^2 plus 6^2 equals 64 plus 36 equals 100.
- So r equals 10 cm.
- Circumference equals 2 pi r equals 2 times 3.142 times 10 equals 62.84, which is 62.8 cm to one decimal place.
Answer: (a) 10 cm (b) 62.8 cm
Where marks slip: B1 for the bisected chord, M1 A1 for Pythagoras, M1 A1 for the circumference. Using 16 instead of 8 in Pythagoras is the standard error and it kills both accuracy marks.
Try one yourself: A chord 24 cm long is 5 cm from the centre. Find the radius. [Answer: 13 cm]
Stuck on a different question?
Paste or photograph it and get the full working, free — no account needed.
Solve my question →Quiz me on this topicQuestions students ask
Do I have to quote circle theorems word for word?
No, but you must give a recognisable reason for each angle when the question says with reasons. Something like angle at centre is twice angle at circumference is enough; a bare number is not.
Can I measure angles from the diagram in the exam?
No. WAEC diagrams are not drawn to scale, and the mark scheme only credits values obtained by reasoning. If your answer could only come from measuring, it scores nothing.
Are these actual WAEC circle geometry past questions?
No, they are original questions in the WASSCE format and difficulty. WAEC owns its past papers, so I write fresh questions that exercise exactly the same theorems the real paper uses.