Circle Geometry Theorems and Formulas: Chords, Tangents, Cyclic Quadrilateral and Common Tangents for SSC

Circle questions in SSC CGL use about ten theorems about angles, chords and tangents. This circle geometry theorems and formulas guide for SSC gives them all in tables with the length formulas for tangents and common tangents, and solved examples.

Angle Theorems

• The angle at the centre is twice the angle at the circumference on the same arc.
• Angles in the same segment are equal.
• The angle in a semicircle is 90°.
• The opposite angles of a cyclic quadrilateral add up to 180°.
• The exterior angle of a cyclic quadrilateral equals the opposite interior angle.
• The angle between a tangent and a chord equals the angle in the alternate segment.

Chord Theorems

• A perpendicular from the centre to a chord bisects the chord.
• Equal chords are at equal distances from the centre.
• Length of a chord at distance d from the centre = 2√(r² − d²).
• Intersecting chords: PA × PB = PC × PD.

Tangent Theorems

Common Tangents of Two Circles

Arc and Sector

Solved Examples

Example 1. The angle at the centre is 100°. Find the angle at the circumference on the same arc.
100/2 = 50°.

Example 2. One angle of a cyclic quadrilateral is 110°. Find the opposite angle.
180 − 110 = 70°.

Example 3. A point is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent.
√(169 − 25) = √144 = 12 cm.

Example 4. Two chords AB and CD intersect at P. PA = 4, PB = 6 and PC = 3. Find PD.
4 × 6 = 3 × PD, so PD = 8.

Example 5. A tangent PT and a secant through P meet the circle at A (nearer) and B. PA = 4 and PB = 9. Find PT.
PT² = 4 × 9 = 36, so PT = 6.

Example 6. The centres of two circles of radii 8 and 3 are 13 cm apart. Find the length of the direct common tangent.
√(169 − 25) = 12 cm. The transverse tangent is √(169 − 121) = 4√3 cm.

Example 7. Find the length of a chord that is 3 cm from the centre of a circle of radius 5 cm.
2√(25 − 9) = 8 cm.

Quick Tips

• Draw the radius to the point of contact when a tangent is given.
• Learn the number of common tangents by the positions of the circles.
• The tangent length formula uses the right triangle with the radius.
• Use PA × PB = PC × PD for chords and secants.

Quick Revision Points

• The angle at the centre is twice the angle at the circumference.
• The angle in a semicircle is 90°.
• The opposite angles of a cyclic quadrilateral add up to 180°.
• Tangent length = √(d² − r²).
• PT² = PA × PB.

Practice Questions With Answers

1. The angle at the centre is 100°. Find the angle at the circumference.
   Answer: 50°
2. What is the angle in a semicircle?
   Answer: 90°
3. One angle of a cyclic quadrilateral is 110°. Find the opposite angle.
   Answer: 70°
4. A point is 13 cm from the centre and the radius is 5 cm. Find the tangent length.
   Answer: 12 cm
5. Chords intersect at P with PA = 4, PB = 6, PC = 3. Find PD.
   Answer: 8
6. PT is a tangent and PA = 4, PB = 9. Find PT.
   Answer: 6
7. Radii are 8 and 3 and the centres are 13 cm apart. Find the direct common tangent.
   Answer: 12 cm
8. How many common tangents do two circles that touch externally have?
   Answer: 3
9. Find the length of a chord that is 3 cm from the centre of a circle of radius 5 cm.
   Answer: 8 cm
10. What is the formula for the length of an arc?
    Answer: (θ/360) × 2πr

How to Practise This Chapter

Write each formula on a card, with a solved example on the back, and revise the cards every day. Then test yourself with our practice papers and the previous year papers page. See all guides on the Formula Guide page, and also read the Mathematics notes and Quantitative Aptitude notes. The previous guide is Triangle Geometry Theorems and the next guide is Data Interpretation.

Frequently Asked Questions

What is the angle at the centre theorem?

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

What is the angle in a semicircle?

It is 90 degrees.

What is a cyclic quadrilateral?

It is a quadrilateral whose four vertices lie on a circle.

What is the sum of the opposite angles of a cyclic quadrilateral?

180 degrees.

What is the length of a tangent from an external point?

√(d² − r²), where d is the distance from the centre.

How many tangents can be drawn from an external point?

Two, and they are equal in length.

How many common tangents do two touching circles have?

Three if they touch externally, and one if they touch internally.

What is the formula for the transverse common tangent?

√(d² − (r1 + r2)²).

Is circle geometry asked in SSC exams?

Yes. Circle questions are asked in SSC CGL maths.

How much time should I take?

About 40 to 60 seconds.

What are common mistakes?

Using the direct tangent formula for the transverse tangent.

How can I learn the theorems?

Draw a figure for each theorem and write the result beside it.