How do I tell a central angle from an angle at the circumference?

Look at the vertex: a vertex at the centre gives a central angle equal to its arc, while a vertex on the circle gives an angle at the circumference, which is half the central angle on the same arc.

Why must a tangent be perpendicular to the radius?

A tangent meets the circle at exactly one point. If the radius were not perpendicular there, the circle would cross the tangent on both sides of that point, so the angle must be 90 degrees.

S3 Mathematics: Circle Properties and Angles at the Circumference (Challenge, 10 Questions)

2026-11-06

Tutor Prep Guide · S3 Mathematics

Ten challenge items needing full reasoning: simultaneous algebraic angles, opposite angles in a ratio, the angle between intersecting chords, choosing the major or minor arc, recovering a distance from a chord, and combining the diameter and tangent theorems, each with a note.

Quick answer Answers in order: 55 degrees, angle A 72 and angle C 108 degrees, 60 degrees, 56 degrees, 5 cm, 5 cm, 130 degrees, 83 degrees, 144 degrees, 38 degrees. When C sits on the minor arc, the angle subtends the major arc.

01 Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	about 45 minutes
Difficulty mix	Challenge 10
Chapter	Chapter: S3 Mathematics: Circle Properties, Angles in a Circle and Tangents
Level	S3

02 Questions (10 items)

ChallengeA central angle on an arc is (4x – 10) degrees and the angle at the circumference on the same arc is (x + 25) degrees. Find that angle at the circumference.
ChallengeIn cyclic quadrilateral ABCD, angle A to angle C is in the ratio 2 : 3. Find both angles.
ChallengeChords AB and CD meet at a point P inside the circle. Arc AC is 80 degrees and arc BD is 40 degrees. Find angle APC.
ChallengePA and PB are tangents from P and angle APB is 68 degrees. Find angle PAB.
ChallengeA circle of radius 13 cm has a chord of length 24 cm. Find the distance from the centre to the chord.
ChallengeAB is a diameter, C lies on the circle, AC = 6 cm and BC = 8 cm. Find the radius.
ChallengeThe central angle AOB is 100 degrees and C lies on the minor arc AB. Find angle ACB.
ChallengeIn cyclic quadrilateral ABCD, angle A is (2x + 15) degrees and angle C is (3x – 5) degrees. Find angle A.
ChallengeTangent PT touches the circle at T and the angle PTA between the tangent and chord TA is 72 degrees. Find the central angle TOA.
ChallengeAB is a diameter, BC is the tangent at B and angle BAC is 38 degrees. Find the angle ABC between the tangent BC and the chord BA.

03 Answers and one-line notes

#	Answer	One-line note
1	55 degrees	Doubling the circumference angle gives 4x – 10 = 2x + 50, so x = 30 and the angle is 55 degrees.
2	angle A is 72 degrees and angle C is 108 degrees	The opposite angles add to 180 degrees over 5 parts, so each part is 36 degrees and the angles are 72 and 108 degrees.
3	60 degrees	The angle between two intersecting chords is half the sum of the intercepted arcs: (80 + 40) / 2 = 60 degrees.
4	56 degrees	Equal tangents make triangle PAB isosceles, so angle PAB is (180 – 68) / 2 = 56 degrees.
5	5 cm	Half the chord is 12 cm, so the distance is the square root of 169 – 144, which is 5 cm.
6	5 cm	Angle ACB is 90 degrees, so AB is the square root of 36 + 64, which is 10 cm, giving a radius of 5 cm.
7	130 degrees	From the minor arc the angle subtends the major arc of 260 degrees, so it is half of that, namely 130 degrees.
8	83 degrees	The opposite angles add to 180 degrees, so 5x + 10 = 180 and x = 34, giving angle A equal to 83 degrees.
9	144 degrees	The tangent-chord angle equals the angle in the alternate segment, and the central angle is twice that, so 72 x 2 = 144 degrees.
10	38 degrees	Angle ACB is 90 degrees so angle ABC inside the triangle is 52 degrees; the tangent is perpendicular to AB, leaving 90 – 52 = 38 degrees.

04 What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.