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CDS Mathematics · Circles

Angles in a Circle

The angle an arc subtends at the centre is twice the angle it subtends at any point on the rest of the circle.

All Circles notesCDS Mathematics strategy

Why this matters

Thirteen PYQs, three of them HARD. Nearly all come down to one rule, centre angle = twice the circumference angle, plus its two consequences: angles in the same segment are equal, and the angle in a semicircle is 90°. The cyclic-quadrilateral rule handles the rest.

Concept 1 of 2: Angle at the centre and angles in the same segment

Fix a chord. Every point on the major arc sees it at the same angle, and the centre sees it at twice that angle. Slide the viewing point along the arc and the angle does not change.

Definition

  • Central angle =2×= 2 \times inscribed angle on the same arc.
  • Angles in the same segment (same side of a chord) are equal.
  • The angle in a semicircle is 90∘90^\circ; so PP is on the circle with diameter ABAB exactly when AP2+BP2=AB2AP^2 + BP^2 = AB^2.
  • An OBTUSE inscribed angle ∠ABC\angle ABC stands on the MAJOR arc ACAC: the reflex angle at the centre is 2∠ABC2\angle ABC, and ∠AOC=360∘−2∠ABC\angle AOC = 360^\circ - 2\angle ABC.
  • Two radii make an isosceles triangle with any chord.

Inscribed angle

∠AOB=2 ∠ACB\angle AOB = 2\,\angle ACB
ABCOinscribed angle at A = half the central angle at O

Worked example

AA, BB, CC lie on a circle with centre OO, and ∠ACB=35∘\angle ACB = 35^\circ with CC on the major arc. Find ∠AOB\angle AOB.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (I) 2026 — Elementary Mathematics · Q52Moderate

Example 1 · Circles · Angles in a Circle

Let O be the centre of a circle. Let A, B and C lie on the circle such that ∠ABC=153∘\angle ABC = 153^\circ. What is the value of ∠AOC\angle AOC ?

An obtuse inscribed angle needs the reflex angle

Doubling an obtuse ∠ABC\angle ABC gives the REFLEX angle at the centre. Doubling 130∘130^\circ to 260∘260^\circ and stopping there answers the wrong angle; ∠AOC\angle AOC is 100∘100^\circ.

Two diameters meet at the centre

If two diameters meet at PP, then PP is the centre, and every segment from PP to the circle is a radius. That makes the triangles isosceles.
Drill 9 more on angle at the centre and angles in the same segment

Concept 2 of 2: Cyclic quadrilaterals and the two segments

The two arcs of a chord add to the whole circle, so the angles seen from them add to half of 360∘360^\circ. That is why opposite angles of a cyclic quadrilateral sum to 180∘180^\circ.

Definition

  • Opposite angles of a cyclic quadrilateral sum to 180∘180^\circ.
  • An exterior angle of a cyclic quadrilateral equals the interior opposite angle.
  • On one chord, the angle in the minor segment and the angle in the major segment are supplementary.
  • The angle in a segment GREATER than a semicircle is acute; in a segment LESS than a semicircle it is obtuse.
  • Equal chords subtend equal angles, so a cyclic quadrilateral with one pair of equal opposite sides has equal diagonals.

Cyclic quadrilateral

∠A+∠C=∠B+∠D=180∘\angle A + \angle C = \angle B + \angle D = 180^\circ

Worked example

A chord is 2\sqrt2 times the radius. Find the angles it subtends at the major arc and at the minor arc.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q84Hard

Example 2 · Circles · Angles in a Circle

The chord of a circle is 3\sqrt{3} times its radius. The angle subtended by this chord at the minor arc is k times the angle subtended at the major arc. What is the value of k ?

Segment, not sector

The textbook rule is about the angle in a SEGMENT. One paper printed 'sector'; a sector bigger than a semicircle has a reflex angle at the centre. Read the word before judging the statement.
Drill 2 more on cyclic quadrilaterals and the two segments

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (3)

Test yourself on Circles

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter testDrill all 13 questions