Why is the angle at the centre always twice the angle at the circumference on the same arc?

Drawing the two radii to the ends of the arc makes two isosceles triangles, and their exterior angles add up to the angle at the centre, exactly twice the angle at the circumference.

Why do opposite angles of a cyclic quadrilateral add to 180 degrees?

All four vertices lie on the circle, so the two arcs opposite a pair of angles make a full turn, the matching central angles total 360 degrees, and each inscribed angle is half of its own arc.

S5 Maths: Circle Geometry Paper 1 (angles at the centre, angles in the same segment and cyclic quadrilaterals)

2026-11-03

S · 5

This paper drills three core theorems: the angle at the centre is twice the angle at the circumference, the angle in a semicircle is a right angle, and opposite angles of a cyclic quadrilateral are supplementary, plus the tangent-radius and tangent-chord results.

Quick answer　40 degrees; 90 and 55 degrees; 65 degrees; 62 and 124 degrees; 120 and 60 degrees; 72 and 108 degrees; 55 and 125 degrees; 78 and 102 degrees; 130 and 25 degrees; the error reverses the doubling between centre and circumference.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	25 minutes
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S5 Maths: Circle Geometry
Level	S5

02　Questions (10 items)

FoundationAn arc subtends 80 degrees at the centre. What angle does it subtend at the circumference?
FoundationAB is a diameter, C lies on the circle and angle BAC is 35 degrees. Find angle ACB and angle ABC.
FoundationABCD is a cyclic quadrilateral with angle A equal to 115 degrees. Find angle C.
FoundationAT is the tangent at T, OT is a radius, and the chord TA makes 62 degrees with the tangent. Find angle OTA and angle AOT.
IntermediateThe angle at the centre exceeds the angle at the circumference on the same arc by 60 degrees. Find both angles.
IntermediateA pair of opposite angles in a cyclic quadrilateral is in the ratio 2 to 3. Find the two angles.
IntermediateChord AB subtends 110 degrees at the centre. C lies on the major arc and D on the minor arc. Find angle ACB and angle ADB.
ChallengeIn cyclic quadrilateral ABCD the exterior angle at A is 78 degrees. Find the interior angle at A and its opposite angle C.
ChallengeTwo tangents from an external point P touch the circle at A and B with angle APB equal to 50 degrees. Find angle AOB, where O is the centre, and angle APO.
Error redoA student writes that the angle at the centre is half the angle at the circumference. What is wrong?

03　Answers and one-line notes

#	Answer	One-line note
1	40 degrees	The angle at the centre is twice the angle at the circumference on the same arc, and 80 over 2 gives 40 degrees.
2	Angle ACB is 90 degrees and angle ABC is 55 degrees	The angle in a semicircle is a right angle, so ACB is 90 degrees, and the angle sum gives 180 minus 90 minus 35, that is 55 degrees.
3	65 degrees	Opposite angles are supplementary, and 180 minus 115 gives 65 degrees.
4	Angle OTA is 62 degrees and angle AOT is 124 degrees	The tangent is perpendicular to the radius, and the tangent-chord angle equals the angle in the alternate segment, 62 degrees, so the angle at the centre is twice that, 124 degrees.
5	The circumference angle is 60 degrees and the centre angle 120 degrees	Let the circumference angle be x, then 2x equals x plus 60, so x is 60 and the centre angle is 120 degrees.
6	72 degrees and 108 degrees	The pair adds to 180 degrees in five parts of 36 degrees each, so two parts make 72 and three parts make 108 degrees.
7	Angle ACB is 55 degrees and angle ADB is 125 degrees	The angle on the major arc is half of 110, that is 55 degrees, and the two angles on opposite sides of the chord are supplementary, so 180 minus 55 is 125 degrees.
8	Angle A is 102 degrees and angle C is 78 degrees	The exterior and interior angles at A are supplementary, so A is 102 degrees, and an exterior angle of a cyclic quadrilateral equals the interior opposite angle, so C is 78 degrees.
9	Angle AOB is 130 degrees and angle APO is 25 degrees	Each tangent is perpendicular to its radius, so the two right angles in OAPB give angle AOB as 180 minus 50, that is 130 degrees, and OP bisects angle APB into 25 degrees.
10	The doubling is reversed: on the same arc the angle at the centre is twice the angle at the circumference, not half of it.	The centre angle is twice the circumference angle on the same arc, and the two are equal only in the degenerate case of 0 degrees, so the half statement never holds.

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.

Find a tutor free Tutor Registration →

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.