SSC CGL Maths · Circles

Angles in a Circle

An arc subtends at the centre twice the angle it subtends at any point on the rest of the circle, and the angle rules for semicircles and cyclic quadrilaterals follow from that one fact.

All Circles notes

Why this matters

Thirty-two questions in the bank, 22 EASY and 10 MODERATE. The EASY ones halve or double a single angle. The MODERATE ones chain two rules, usually a right angle from a diameter with the angle sum of a triangle, or opposite angles of a cyclic quadrilateral.

Concept 1 of 4: Angle at the centre is twice the angle at the circumference

Look at an arc AB from the centre O and from a point C on the rest of the circle. From the centre the arc looks twice as wide. Every point C on the same remaining arc sees it at the same angle, because the centre angle does not change.

Definition

Angle at the centre $= 2 \times$ angle at the circumference, when both stand on the same arc and C is on the remaining part of the circle.
Angles in the same segment are equal: a chord subtends the same angle at every point of one arc.
A chord whose centre angle is $\theta$ subtends $\frac{\theta}{2}$ at a point of the major arc. A point of the minor arc stands on the major arc, so it sees half the reflex angle: $\frac{360^\circ - \theta}{2} = 180^\circ - \frac{\theta}{2}$ .
The reflex angle at the centre is $360^\circ - \theta$ . An arc's degree measure is its centre angle; a semicircle is $180^\circ$ .
Two radii make an isosceles triangle: in triangle OAB, $OA = OB$ , so $\angle OAB = \angle OBA = \frac{180^\circ - \angle AOB}{2}$ .
Arcs given in a ratio share $360^\circ$ (or $180^\circ$ on one side of a diameter) in that ratio.

Centre and circumference

\angle AOB = 2\,\angle ACB \qquad \text{reflex } \angle AOB = 360^\circ - \angle AOB

Othe centre
Ca point on the arc that does not contain the arc AB

Worked example

A chord AB of a circle subtends 124° at the centre O. Find the angle AB subtends at a point C on the major arc and at a point D on the minor arc.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 3 · Q73Easy

Example 1 · Circles · Angles in a Circle

A chord subtends an angle of

70^\circ

at the center of a circle. What is the measure of the angle subtended by the same chord at any point on the remaining part of the circle?

Doubling instead of halving

The angle at the centre is the larger one. From the centre angle you halve to reach the circumference; from a circumference angle you double to reach the centre.

Halving the centre angle for a point on the minor arc

A point on the minor arc sees half the reflex angle, which is 180° minus half the centre angle. For a centre angle of 100°, that point sees 130°, not 50°.

Drill 16 more on angle at the centre is twice the angle at the circumference

Concept 2 of 4: Angle in a semicircle and the circle through three points

A diameter is a chord whose centre angle is a straight angle, 180°. Half of that is 90°, so every point on the circle sees a diameter at a right angle. Turned round, the longest side of any right triangle is a diameter of the circle through its corners.

Definition

The angle in a semicircle is $90^\circ$ : if AB is a diameter and C is on the circle, $\angle ACB = 90^\circ$ .
So the hypotenuse of a right triangle is a diameter of its circumcircle, and the circumradius is half the hypotenuse.
Exactly one circle passes through three points that are not in a straight line. No circle passes through three points in a line. If two of the three distances add up to the third, the points are in a line.
Two circles meet at P and Q, with diameters PR and PS. Then $\angle PQR = \angle PQS = 90^\circ$ , so R, Q and S lie in one line.
When a diameter appears in a triangle question, mark the $90^\circ$ first, then use the angle sum of the triangle.

Diameter and right angle

\angle ACB = 90^\circ \text{ when } AB \text{ is a diameter} \qquad R = \frac{\text{hypotenuse}}{2}

Rthe radius of the circle through the three corners

Worked example

AB is a diameter of a circle and C is a point on the circle with

\angle CAB = 34^\circ

. Find

\angle ABC

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 1 · Q71Moderate

Example 2 · Circles · Angles in a Circle

In a cyclic quadrilateral ABCD, AB is the diameter of the circle. If angle ACD is

40^\circ

, what is the measure of angle BAD?

Taking a shorter side of a right triangle as the diameter

Only the hypotenuse, the side facing the right angle, is a diameter of the circle through the three corners.

Assuming any three points lie on a circle

Three points have a circle through them only when they are not in a line. If two of the distances add up to the third, the answer is zero circles.

Drill 5 more on angle in a semicircle and the circle through three points

Concept 3 of 4: Opposite angles of a cyclic quadrilateral

Put four points on a circle and join them in order. Two opposite corners stand on the two arcs that make up the whole circle. Their centre angles add to 360°, so the corners themselves add to half of that, 180°.

Definition

A cyclic quadrilateral has all four corners on one circle.
Opposite angles add to $180^\circ$ : $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$ . Adjacent angles follow no such rule.
An exterior angle equals the interior opposite angle. If side QR is produced beyond R to T, then $\angle SRT = \angle SPQ$ .
With the centre O: if $\angle QOS = \theta$ faces the corner R, then $\angle QPS = \frac{\theta}{2}$ and $\angle QRS = 180^\circ - \frac{\theta}{2}$ .
A diameter as one side gives a right angle at each of the other two corners, which often finishes the question.

Cyclic quadrilateral

\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ

Worked example

ABCD is a cyclic quadrilateral with

\angle A = (4x + 10)^\circ

and

\angle C = (2x + 20)^\circ

. Find

\angle A

and

\angle C

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 19 Sep 2024, Shift 2 · Q52Moderate

Example 3 · Circles · Angles in a Circle

A quadrilateral PQRS is inscribed in a circle of centre O, such that PQ is a diameter and

\angle PSR = 120^\circ

. Find the value of

\angle QPR

Adding adjacent angles to 180°

Only opposite angles of a cyclic quadrilateral add to 180°. Two neighbouring angles can add to anything.

Matching an exterior angle with the wrong interior angle

An exterior angle of a cyclic quadrilateral equals the interior angle at the opposite corner, not the interior angle beside it.

Drill 4 more on opposite angles of a cyclic quadrilateral

Concept 4 of 4: A chord's length from the angle it subtends

The two radii to the ends of a chord and the chord itself form an isosceles triangle. Its top angle is the centre angle, so that angle fixes the chord. For 60°, 90° and 120° the triangle is one you already know.

Definition

Triangle OAB has $OA = OB = r$ and top angle $\theta$ , the centre angle. An angle at the circumference must be doubled first.
$\theta = 60^\circ$ : the triangle is equilateral, so the chord equals the radius. (A chord subtending $30^\circ$ on the major arc is a radius long.)
$\theta = 90^\circ$ : chord $= r\sqrt{2}$ . $\theta = 120^\circ$ : chord $= r\sqrt{3}$ . $\theta = 180^\circ$ : chord $= 2r$ .
Drop the perpendicular from O: it halves both the chord and the angle. A $30^\circ$ , $60^\circ$ , $90^\circ$ triangle has sides in the ratio $1 : \sqrt{3} : 2$ , and a $45^\circ$ , $45^\circ$ , $90^\circ$ triangle $1 : 1 : \sqrt{2}$ .
In general the chord is $2r\sin\frac{\theta}{2}$ and its distance from the centre is $r\cos\frac{\theta}{2}$ .
In a triangle on a diameter, a side equal to the radius faces $30^\circ$ , so the angles are $30^\circ$ , $60^\circ$ and $90^\circ$ .

Chord from the centre angle

c = 2r\sin\frac{\theta}{2}

cthe chord
$\theta$ the angle the chord subtends at the centre

Worked example

A chord of a circle of radius 9 cm subtends 120° at the centre. Find its length and its distance from the centre.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 16 Sep 2025, Shift 1 · Q74Moderate

Example 4 · Circles · Angles in a Circle

A chord of a circle has a length of 12 cm. The angle subtended by the chord at a point on the circumference is

30^\circ

. What is the distance from the center of the circle to the chord?

Using the angle at the circumference as the centre angle

A chord that subtends 30° at a point on the circle subtends 60° at the centre. Double first, then use the triangle of two radii.

Drill 3 more on a chord's length from the angle it subtends

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 (4)

Angle at the centre is twice the angle at the circumference
Centre and circumference
$\angle AOB = 2\,\angle ACB \qquad \text{reflex } \angle AOB = 360^\circ - \angle AOB$
Angle in a semicircle and the circle through three points
Diameter and right angle
$\angle ACB = 90^\circ \text{ when } AB \text{ is a diameter} \qquad R = \frac{\text{hypotenuse}}{2}$
Opposite angles of a cyclic quadrilateral
Cyclic quadrilateral
$\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ$
A chord's length from the angle it subtends
Chord from the centre angle
$c = 2r\sin\frac{\theta}{2}$

Watch out for (7)

Doubling instead of halving
→ Angle at the centre is twice the angle at the circumference
Halving the centre angle for a point on the minor arc
→ Angle at the centre is twice the angle at the circumference
Taking a shorter side of a right triangle as the diameter
→ Angle in a semicircle and the circle through three points
Assuming any three points lie on a circle
→ Angle in a semicircle and the circle through three points
Adding adjacent angles to 180°
→ Opposite angles of a cyclic quadrilateral
Matching an exterior angle with the wrong interior angle
→ Opposite angles of a cyclic quadrilateral
Using the angle at the circumference as the centre angle
→ A chord's length from the angle it subtends

Test yourself on Circles

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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