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Chapter 11 · Question 3

Derive the formula for the area of a segment of a circle. Write the expression for the area of the minor segment APBAPB of a circle with centre OO, radius rr, and sector angle θ\theta.

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Answer

Direct Answer

The area of the minor segment is the area of the corresponding minor sector minus the area of the triangle formed by the two radii and the chord. For θ\theta measured in degrees, Area of minor segment APB=θ360×πr2−12r2sin⁡θ\text{Area of minor segment } APB = \frac{\theta}{360} \times \pi r^2 - \frac{1}{2}r^2\sin\theta.

Simple Explanation

To find the area of a segment (the 'cap' of the circle), subtract the area of the triangle from the area of the pizza-slice (sector) that contains it. For the major segment, subtract the minor segment area from the whole circle.

Exam-Ready Structure

Consider a circle with centre OO, radius rr, and a minor sector OAPBOAPB of angle θ\theta. The chord ABAB divides the minor sector into the triangle △OAB\triangle OAB and the minor segment APBAPB. Therefore, for θ\theta measured in degrees: Area of minor segment APB=θ360×πr2−12r2sin⁡θ.\text{Area of minor segment } APB = \frac{\theta}{360} \times \pi r^2 - \frac{1}{2}r^2\sin\theta. For the major segment AQBAQB: Area of major segment AQB=πr2−Area of minor segment APB.\text{Area of major segment } AQB = \pi r^2 - \text{Area of minor segment } APB. For specific angles, the same triangle formula can be evaluated directly or by splitting the isosceles triangle into right triangles.

Key Points

  • Area of minor segment = Area of sector − Area of triangle OAB
  • Sector area = (θ/360) × πr²
  • Area of major segment = πr² − area of minor segment
  • Triangle OAB has area 12r2sin⁡θ\frac{1}{2}r^2\sin\theta when θ\theta is measured in degrees
  • For θ=60°: triangle is equilateral; for θ=90°: right isosceles

Related Questions

Q2

Derive the formulas for (i) the area of a sector of angle θ\theta and radius rr, and (ii) the length of the arc of a sector of angle θ\theta and radius rr.

Q4

Find the area of the sector of a circle with radius 4 cm4\text{ cm} and of angle 30∘30^\circ. Also, find the area of the corresponding major sector. (Use π=3.14\pi = 3.14)

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