Circle Geometry Notes for SSC Exams

Circle geometry is the second most important geometry topic in SSC CGL after triangles. Two to three questions come from circles in each Tier 1 paper, mostly on tangents, chords and angles. These circle geometry notes for SSC give you the theorems in a quick-revision format, with formulas, solved examples and practice questions.

Basic Terms

• Radius: distance from the centre to the circle. Diameter: twice the radius.
• Chord: a line joining two points on the circle. The diameter is the longest chord.
• Tangent: a line touching the circle at exactly one point.
• Secant: a line cutting the circle at two points.
• Arc: a part of the circle. Segment: region between a chord and an arc. Sector: region between two radii and an arc.

Chord Properties

• The perpendicular from the centre to a chord bisects the chord.
• Equal chords are equally distant from the centre.
• If a chord of length 2x is at distance d from the centre, then r squared = x squared + d squared.
• The longer the chord, the closer it is to the centre.

Tangent Properties

• A tangent is perpendicular to the radius at the point of contact.
• Two tangents drawn from an external point are equal in length.
• Length of a tangent from a point at distance d from the centre = square root of (d squared - r squared).
• Direct common tangent between two circles = square root of (d squared - (r1 - r2) squared).
• Transverse common tangent = square root of (d squared - (r1 + r2) squared).
• Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Angle Theorems

Solved Examples

Example 1: The angle at the centre of a circle is 100 degrees. Find the angle at the circumference on the same arc.
Solution: Angle at the circumference = 100 / 2 = 50 degrees.

Example 2: ABCD is a cyclic quadrilateral and angle A = 80 degrees. Find angle C.
Solution: Opposite angles add up to 180 degrees, so angle C = 100 degrees.

Example 3: A point is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from the point.
Solution: Length = square root of (169 - 25) = square root of 144 = 12 cm.

Example 4: A chord of length 16 cm is in a circle of radius 10 cm. Find its distance from the centre.
Solution: Half chord = 8. Distance = square root of (100 - 64) = 6 cm.

Example 5: Find the length of the direct common tangent between two circles with radii 8 cm and 3 cm whose centres are 13 cm apart.
Solution: Length = square root of (169 - 25) = 12 cm.

Example 6: What is the angle subtended by a diameter at any point on the circle?
Solution: The angle in a semicircle is 90 degrees.

Quick Tips to Solve Faster

• Always join the centre to the point of contact to use the right angle with the tangent.
• Draw the perpendicular from the centre to the chord to form a right triangle.
• Remember the Pythagorean triplets to avoid square roots in chord and tangent problems.
• Use the cyclic quadrilateral rule whenever four points lie on a circle.
• In common tangent questions, identify the type (direct or transverse) before using the formula.

Common Mistakes to Avoid

• Using the diameter in place of the radius in the tangent formula.
• Taking the angle at the circumference as equal to the central angle.
• Mixing the formulas for direct and transverse common tangents.
• Forgetting that opposite angles of a cyclic quadrilateral add up to 180 degrees.
• Not halving the chord while using the distance formula.

Quick Revision Points

• Angle at the centre = 2 x angle at the circumference.
• Tangent is perpendicular to the radius.
• Tangent length = square root of (d squared - r squared).
• Cyclic quadrilateral: opposite angles sum to 180 degrees.
• Direct common tangent uses (r1 - r2) and transverse uses (r1 + r2).

Practice Questions With Answers

1. The angle at the circumference is 35 degrees. Find the angle at the centre on the same arc.
   Answer: 70 degrees (2 x 35).
2. ABCD is a cyclic quadrilateral and angle B = 105 degrees. Find angle D.
   Answer: 75 degrees (180 - 105).
3. A point is 17 cm from the centre of a circle of radius 8 cm. Find the tangent length.
   Answer: 15 cm (Triplet 8-15-17).
4. A chord of length 24 cm is in a circle of radius 13 cm. Find its distance from the centre.
   Answer: 5 cm (Triplet 5-12-13).
5. What is the angle in a semicircle?
   Answer: 90 degrees (Standard result).
6. The diameter of a circle is 26 cm and a chord is 5 cm from the centre. Find the chord length.
   Answer: 24 cm (2 x square root of (169 - 25)).
7. What is the angle between a tangent and the radius at the point of contact?
   Answer: 90 degrees (Standard result).
8. Two circles of radii 7 cm and 1 cm have centres 10 cm apart. Find the length of the direct common tangent.
   Answer: 8 cm (Square root of (100 - 36)).
9. The angle between two tangents from an external point is 60 degrees. Find the angle subtended by the chord of contact at the centre.
   Answer: 120 degrees (180 - 60).
10. The hypotenuse of a right triangle is 10 cm. Find its circumradius.
    Answer: 5 cm (Hypotenuse / 2).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching practice papers to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Triangle Geometry Notes
• Mensuration 2D Notes
• Trigonometry Notes
• Heights and Distances Notes
• SSC Mensuration Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is a tangent to a circle?

A tangent is a line that touches the circle at exactly one point, called the point of contact.

What is the angle between a tangent and the radius?

It is always 90 degrees at the point of contact.

What is the relation between the central angle and the angle at the circumference?

The angle at the centre is twice the angle at the circumference on the same arc.

What is a cyclic quadrilateral?

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

What is the formula for the tangent length from an external point?

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

How many questions come from circles in SSC CGL Tier 1?

Two or three questions come from circle geometry in most papers.

What is the angle in a semicircle?

It is always 90 degrees.

How should I practise circle geometry?

Learn the theorems, draw each figure and solve the examples here before practising previous year questions.