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Geometry for SSC CGL: circles

Circle questions in SSC CGL test a compact set of theorems — angles in the same segment, the angle at the centre, cyclic quadrilaterals, tangents and the alternate segment theorem, chords, and common tangents between two circles. Learn each with a diagram in mind and most questions take under a minute. The theorems, worked examples and practice with answers.

4 Oct 2026 4 min read

In this guide
  1. Chords
  2. Angles
  3. Cyclic quadrilaterals
  4. Tangents
  5. Intersecting chords and secants
  6. Two circles
  7. Incircle of a right triangle
  8. Common traps
  9. Practice

A circle question in SSC CGL often looks like a tangle of lines — chords crossing, tangents meeting outside, a quadrilateral drawn inside. But almost every one rests on one of about ten theorems. The trick is to recognise which one applies: is there an angle at the centre? A cyclic quadrilateral? A tangent meeting a radius? This post collects those theorems with worked examples.

Chords

  • The perpendicular from the centre to a chord bisects the chord.
  • Equal chords are equidistant from the centre.
  • If a chord of length 2l is at distance d from the centre of a circle of radius r: r² = l² + d².

Worked example: A chord of length 16 cm is 6 cm from the centre. Find the radius.
r² = 8² + 6² = 100 → r = 10 cm.

Angles

TheoremStatement
Angle at the centreThe angle subtended at the centre is twice the angle at the circumference by the same arc
Same segmentAngles in the same segment are equal
SemicircleThe angle in a semicircle is 90°

Worked example: An arc subtends 110° at the centre. Find the angle it subtends at a point on the remaining part of the circle.
110/2 = 55°.

Cyclic quadrilaterals

A quadrilateral whose vertices lie on a circle:

  • Opposite angles sum to 180°.
  • An exterior angle equals the interior opposite angle.

Worked example: In cyclic quadrilateral ABCD, ∠A = 3x and ∠C = 2x. Find x.
3x + 2x = 180 → x = 36.

Tangents

  • A tangent is perpendicular to the radius at the point of contact.
  • The two tangents from an external point are equal in length.
  • Tangent length from a point at distance d from the centre: √(d² − r²).

Worked example: From a point 13 cm from the centre of a circle of radius 5 cm, find the length of the tangent.
√(169 − 25) = √144 = 12 cm.

Alternate segment theorem

The angle between a tangent and a chord equals the angle in the alternate segment.

Intersecting chords and secants

  • Chords intersecting inside: if chords AB and CD meet at P, then PA × PB = PC × PD.
  • Secants from an outside point: PA × PB = PC × PD, with each measured from P.
  • Tangent and secant: PT² = PA × PB.

Worked example: From point P, a tangent PT and a secant PAB are drawn. PA = 4 cm and AB = 5 cm. Find PT.
PB = 9. PT² = 4 × 9 = 36 → PT = 6 cm.

Two circles

For two circles with radii r₁ and r₂ and centres d apart:

TangentLength
Direct common tangent√[d² − (r₁ − r₂)²]
Transverse common tangent√[d² − (r₁ + r₂)²]

Worked example: Two circles of radii 8 cm and 3 cm have centres 13 cm apart. Find the length of the direct common tangent.
√(169 − 25) = 12 cm.

Number of common tangents:

PositionCommon tangents
Far apart (d > r₁ + r₂)4
Touching externally3
Intersecting2
Touching internally1
One inside the other0

Incircle of a right triangle

For a right triangle with legs a and b and hypotenuse c, the inradius is r = (a + b − c)/2.

Worked example: Legs 6 and 8, hypotenuse 10: r = (6 + 8 − 10)/2 = 2.

Common traps

TrapCorrect approach
Halving the wrong angleThe centre angle is double the circumference angle
Forgetting the right angle between tangent and radiusMark it on your figure first
Confusing direct and transverse formulasDirect: difference of radii; transverse: sum

Practice

  1. Find the length of a chord that is 5 cm from the centre of a circle of radius 13 cm.
  2. In a cyclic quadrilateral, one angle is 75°. Find the opposite angle.
  3. Two tangents are drawn to a circle from an external point and the angle between them is 60°. Find the angle subtended by the chord of contact at the centre.
  4. Chords AB and CD intersect at P inside a circle. PA = 6, PB = 4 and PC = 3. Find PD.
  5. Two circles of radii 5 and 3 have centres 10 cm apart. Find the length of the transverse common tangent.
  6. Find the inradius of a right triangle with sides 5, 12 and 13.

Answers: 1. 24 cm. 2. 105°. 3. 120°. 4. 8. 5. 6 cm. 6. 2.

A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .

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