SSC Circle Questions with Solutions (Chords, Tangents and Angles)

Circles are one of the most important parts of SSC geometry. Questions come from chords, tangents, angles in a circle, cyclic quadrilaterals and common tangents. Once you know the theorems, most questions can be solved in 30 to 40 seconds. This practice paper gives you 21 circle questions with answers and short solutions.

Practice Paper at a Glance

Basic Circle Terms in Simple Words

A circle is a set of all points at the same distance (radius) from a fixed point (centre).

• Chord: a line joining two points of the circle. The longest chord is the diameter.
• Tangent: a line that touches the circle at only one point.
• Secant: a line that cuts the circle at two points.
• Arc and segment: a part of the circle and the region cut by a chord.

Important Theorems and Shortcut Tricks

• The angle at the centre is twice the angle at the circumference on the same arc.
• The angle in a semicircle is 90 degrees, and angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral add up to 180 degrees.
• A tangent is perpendicular to the radius at the point of contact, and tangents from an external point are equal.
• Intersecting chords: PA x PB = PC x PD. Tangent and secant: PT^2 = PA x PB.
• The perpendicular from the centre to a chord bisects the chord: r^2 = d^2 + (chord/2)^2.
• Common tangents: direct = square root of (d^2 - (R - r)^2) and transverse = square root of (d^2 - (R + r)^2).
• Circles touching externally: d = R + r. Touching internally: d = R - r.

Set 1: Basic Theorems (Q1–Q7)

Q1. A chord of length 16 cm is in a circle of radius 10 cm. Find its distance from the centre.
Answer: 6 cm
Solution: Half chord = 8. Distance = square root of (100 - 64) = 6 cm.

Q2. The angle at the centre of a circle is 100 degrees. Find the angle at the circumference on the same arc.
Answer: 50 degrees
Trick: The angle at the circumference is half of the angle at the centre.

Q3. What is the angle in a semicircle?
Answer: 90 degrees
Solution: It is always a right angle.

Q4. One angle of a cyclic quadrilateral is 70 degrees. Find the opposite angle.
Answer: 110 degrees
Solution: Opposite angles add up to 180 degrees.

Q5. A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P.
Answer: 12 cm
Solution: Tangent = square root of (13^2 - 5^2) = 12 cm.

Q6. PA and PB are tangents from P to a circle. If PA = 8 cm, find PB.
Answer: 8 cm
Solution: Tangents from an external point are equal.

Q7. What is the angle between a tangent and the radius at the point of contact?
Answer: 90 degrees
Solution: The tangent is perpendicular to the radius.

Set 2: Chords, Tangents and Angles (Q8–Q14)

Q8. Two chords AB and CD intersect at P inside a circle. If PA = 4, PB = 6 and PC = 3, find PD.
Answer: 8
Trick: PA x PB = PC x PD, so 24 = 3 x PD and PD = 8.

Q9. From an external point P, a secant PAB and a tangent PT are drawn. If PA = 4 and AB = 5, find PT.
Answer: 6
Trick: PB = 9. PT^2 = PA x PB = 36, so PT = 6.

Q10. The angle between a tangent and a chord at the point of contact is 40 degrees. Find the angle in the alternate segment.
Answer: 40 degrees
Solution: By the alternate segment theorem, the angles are equal.

Q11. An arc subtends 120 degrees at the centre. Find the angle it subtends at a point on the remaining circumference.
Answer: 60 degrees
Solution: It is half of the central angle.

Q12. A chord of length 24 cm is at a distance of 5 cm from the centre. Find the radius.
Answer: 13 cm
Solution: r = square root of (12^2 + 5^2) = 13 cm.

Q13. PA and PB are tangents from P to a circle with centre O. If angle APB = 70 degrees, find angle AOB.
Answer: 110 degrees
Solution: Angle AOB + angle APB = 180 degrees.

Q14. In a cyclic quadrilateral ABCD, angle A = 3x and angle C = 2x + 20. Find angle A.
Answer: 96 degrees
Solution: 3x + 2x + 20 = 180, so x = 32 and angle A = 96 degrees.

Set 3: Exam-Level Questions (Q15–Q21)

Q15. Two circles of radii 8 cm and 3 cm have their centres 13 cm apart. Find the length of the direct common tangent.
Answer: 12 cm
Trick: Length = square root of (13^2 - (8 - 3)^2) = square root of 144 = 12 cm.

Q16. Two circles of radii 6 cm and 4 cm touch each other externally. Find the distance between their centres.
Answer: 10 cm
Solution: d = R + r = 10 cm.

Q17. Two circles of radii 9 cm and 4 cm touch each other internally. Find the distance between their centres.
Answer: 5 cm
Solution: d = R - r = 5 cm.

Q18. Find the radius of the incircle of a triangle with sides 6 cm, 8 cm and 10 cm.
Answer: 2 cm
Solution: Area = 24 and s = 12, so r = area / s = 2 cm.

Q19. Find the radius of the circumcircle of a right triangle with sides 6 cm, 8 cm and 10 cm.
Answer: 5 cm
Trick: The circumradius of a right triangle is half the hypotenuse.

Q20. Find the length of an arc that subtends 90 degrees at the centre of a circle of radius 14 cm.
Answer: 22 cm
Solution: Arc = (90 / 360) x 2 x 22/7 x 14 = 22 cm.

Q21. Angle ACB = 35 degrees in a circle. Find angle ADB if C and D are on the same side of AB.
Answer: 35 degrees
Solution: Angles in the same segment are equal.

Common Mistakes in Circle Questions

• Using the angle at the centre as equal to the angle at the circumference (it is double).
• Forgetting that the perpendicular from the centre bisects the chord.
• Using the direct common tangent formula for the transverse tangent.
• Adding the opposite angles of a cyclic quadrilateral wrongly.
• Mixing up PA x PB = PT^2 with PA x PB = PC x PD.

How to Practice These 21 Questions

1. Make a one-page sheet of the circle theorems.
2. Draw a figure for every question.
3. Solve each set in 8 minutes.
4. Practise tangent and chord questions separately.
5. Revise the mistakes after one day.

Related Topics to Study Next

• Mensuration Questions
• Trigonometry Questions
• SSC Algebra Questions
• Fractions and Decimals Questions

Frequently Asked Questions

What is a tangent to a circle?

A tangent is a line that touches the circle at exactly one point and is perpendicular to the radius at that point.

What is a chord?

A chord is a line segment joining two points on the circle. The diameter is the longest chord.

What is the angle at the centre theorem?

The angle subtended by an arc at the centre is twice the angle subtended at any point on the remaining circumference.

What is the angle in a semicircle?

The angle in a semicircle is always 90 degrees.

What is a cyclic quadrilateral?

It is a quadrilateral whose all four vertices lie on a circle. Its opposite angles add up to 180 degrees.

What is the alternate segment theorem?

The angle between a tangent and a chord is equal to the angle in the alternate segment.

How do you find the length of a tangent from an external point?

Tangent length = square root of (d^2 - r^2), where d is the distance from the centre.

What is the formula for intersecting chords?

If two chords intersect at P, then PA x PB = PC x PD.

What is the tangent-secant theorem?

If a tangent PT and a secant PAB are drawn from P, then PT^2 = PA x PB.

How many circle questions come in SSC exams?

Usually two to three questions per shift come from circles and related theorems in CGL.

What is the formula for the common tangents of two circles?

Direct common tangent = square root of (d^2 - (R - r)^2). Transverse common tangent = square root of (d^2 - (R + r)^2).

How should I prepare circles for SSC exams?

Learn the theorems, draw clear figures, solve topic-wise questions and revise previous year geometry questions.