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 (Foundation, 10 Questions)

2026-11-06

Tutor Prep Guide · S3 Mathematics

Ten foundation items on circle vocabulary and three core theorems: radius against diameter, the angle at the circumference being half the central angle on the same arc, the angle in a semicircle, and the tangent perpendicular to the radius. Each answer states the theorem before substituting numbers.

Quick answer Answers in order: 14 cm, 40 degrees, 90 degrees, 90 degrees, 70 degrees, 13 cm, 48 degrees, 58 degrees, 1/3, 9 cm. The two most used facts are that angles on the same arc are equal and half the central angle.

01 Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	about 25 minutes
Difficulty mix	Foundation 10
Chapter	Chapter: S3 Mathematics: Circle Properties, Angles in a Circle and Tangents
Level	S3

02 Questions (10 items)

FoundationThe radius of a circle is 7 cm. Find its diameter.
FoundationA central angle standing on an arc is 80 degrees. Find the angle at the circumference standing on the same arc.
FoundationAB is a diameter and C lies on the circle. Find angle ACB.
FoundationPA is a tangent touching the circle at A and O is the centre. Find angle OAP.
FoundationAn angle at the circumference standing on an arc is 35 degrees. Find the central angle on the same arc.
FoundationThe diameter of a circle is 26 cm. Find its radius.
FoundationTwo angles at the circumference stand on the same arc and one of them is 48 degrees. Find the other.
FoundationAB is a diameter and C lies on the circle. If angle ABC is 32 degrees, find angle BAC.
FoundationA central angle is 120 degrees. What fraction of the whole circumference is its arc?
FoundationTwo tangents from an external point P touch the circle at A and B. If PA = 9 cm, find PB.

03 Answers and one-line notes

#	Answer	One-line note
1	14 cm	The diameter is twice the radius, so 7 x 2 = 14 cm.
2	40 degrees	The angle at the circumference is half the central angle on the same arc: 80 / 2 = 40 degrees.
3	90 degrees	The angle in a semicircle is a right angle, so angle ACB is 90 degrees.
4	90 degrees	A radius is perpendicular to the tangent at the point of contact, so angle OAP is 90 degrees.
5	70 degrees	The central angle is twice the angle at the circumference: 35 x 2 = 70 degrees.
6	13 cm	The radius is half the diameter, so 26 / 2 = 13 cm.
7	48 degrees	Angles at the circumference on the same arc are equal, so the other is also 48 degrees.
8	58 degrees	Angle ACB is 90 degrees, so angle BAC is 180 – 90 – 32 = 58 degrees.
9	1/3	A full turn is 360 degrees, and 120 / 360 reduces to 1/3.
10	9 cm	Tangents from the same external point are equal in length, so PB = PA = 9 cm.

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.