Circles and Their Properties – Complete Notes for Competitive Exams

1. Introduction

A circle is a set of all points in a plane that are at a fixed distance (radius) from a fixed point called the center.

Circles are a key topic in Geometry for SSC, Banking, Railways, and other competitive exams.

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2. Important Terminology

• Center (O): Fixed point inside the circle
• Radius (r): Distance from center to any point on the circle
• Diameter (d): $d = 2r$d=2r
• Chord: Line joining two points on the circle
• Tangent: Line touching the circle at exactly one point
• Secant: Line intersecting the circle at two points
• Arc: Part of the circumference
• Sector: Region enclosed by two radii and an arc
• Segment: Region enclosed by a chord and an arc

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3. Important Formulas

3.1 Circle Geometry

• Circumference: $C = 2\pi r$C=2πr
• Area: $A = \pi r^2$A=πr2
• Length of arc: $l = \frac{\theta}{360} \cdot 2\pi r$l=360θ​⋅2πr
• Area of sector: $A_s = \frac{\theta}{360} \cdot \pi r^2$As​=360θ​⋅πr2
• Area of segment: $A_{seg} = \text{Area of sector} – \text{Area of triangle}$Aseg​=Area of sector−Area of triangle

3.2 Tangent Properties

1. Tangent at any point is perpendicular to the radius
2. Length of tangent from external point P: $PT = \sqrt{PO^2 – r^2}$PT=PO2−r2​

3.3 Chord Properties

1. Chords equidistant from center are equal
2. Perpendicular from center to chord bisects the chord

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4. Important Tips

• Draw diagram first; most circle problems are visual
• Remember Tangent ⊥ Radius property
• Use sector and segment formulas carefully
• For chords, use perpendicular bisector property

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Top 25 Practice Questions – Circles

Basic Properties & Geometry

Q1. Find circumference of a circle with radius 7 cm
Q2. Find area of a circle with diameter 14 cm
Q3. Find the radius of a circle with circumference 44 cm
Q4. Length of an arc of circle with r = 14 cm, θ = 60°
Q5. Area of sector with r = 10 cm, θ = 90°
Q6. Find area of segment if r = 7 cm, θ = 60°
Q7. Chord length of circle with r = 5 cm, distance from center = 3 cm
Q8. If diameter = 20 cm, find area and circumference
Q9. Length of tangent from external point P at distance 13 cm from center r = 5 cm
Q10. If two chords are 6 cm and 8 cm away from center, which is longer?

Tangents & Angles

Q11. Prove that tangent at any point is perpendicular to radius
Q12. Length of tangent from external point at distance 10 cm from circle center r = 6 cm
Q13. Angle between two tangents drawn from external point 20 cm away from circle radius 12 cm
Q14. Distance between two parallel tangents of circle with r = 7 cm
Q15. Two tangents form angle 60° at external point. Find radius if distance from point to center = 10 cm

Advanced Questions

Q16. Area of segment formed by chord 8 cm in circle with r = 5 cm
Q17. Find radius of circle if area of sector θ = 90°, sector area = 50 cm²
Q18. Two chords intersect at right angle inside circle, find distance from center
Q19. Find chord length subtending 60° at center, r = 7 cm
Q20. Find distance between two chords of equal length 6 cm in circle r = 5 cm
Q21. Area of shaded region between two concentric circles r1 = 14 cm, r2 = 7 cm
Q22. If tangent touches circle at point A, find angle between radius and tangent at A
Q23. Circle inscribed in square with side 14 cm, find area of circle
Q24. Circle circumscribes square with side 10 cm, find radius of circle
Q25. Sector of circle r = 7 cm subtends 120° at center. Find perimeter of sector

Answer

Answers – Circles

Q1. 44 cm
Q2. 154 cm²
Q3. 7 cm
Q4. 14.66 cm
Q5. 78.54 cm²
Q6. 7.7 cm² (approx)
Q7. 8 cm
Q8. Area = 314.16 cm², Circumference = 62.83 cm
Q9. 12 cm
Q10. 8 cm chord is longer
Q11. Proven by perpendicular property
Q12. 8 cm
Q13. 2 × radius × sin(θ/2) method → 12 cm (approx)
Q14. 14 cm
Q15. r = 5 cm
Q16. 4.67 cm² (approx)
Q17. r = 10.6 cm (approx)
Q18. Depends on triangle method → 3.54 cm (example)
Q19. 7 cm
Q20. 6 cm
Q21. 461.81 cm²
Q22. 90°
Q23. 154 cm²
Q24. r = 7.07 cm
Q25. Perimeter = 38.59 cm (approx)