Question

The radius of a circle with centre O is 5 cm (Figure). Two radii OA and OB are drawn at right angles to each other. Find the areas of the segments made by the chord AB (Take π = 3.14).

Answer 1

Given:

Radius OA = OB = 5 cm, OA ⟂ OB ⇒ central angle ∠AOB = 90°.

Take π = 3.14. Method: minor segment area = area of sector AOB – area of ΔAOB; major segment area = area of circle – minor segment.

Step-wise calculation:

1. Area of circle = πr² 

= 3.14 × 25

= 78.50 cm²

2. Area of sector AOB (θ = 90°)

= `(θ/360) xx πr^2` 

= `(90/360) xx 78.50`

= `1/4 xx 78.50`

= 19.625 cm²

3. Area of ΔAOB (two radii with included angle 90°)

= `(1/2)·OA·OB·sin90^circ` 

= `(1/2)·5·5·1`

= 12.50 cm²

4. Area of minor segment (closer to the chord AB)

= Sector – Triangle

= 19.625 – 12.50

= 7.125 cm²

5. Area of major segment

= Area of circle – Minor segment 

= 78.50 – 7.125

= 71.375 cm²

Minor segment area = 7.125 cm² (≈ 7.13 cm²).

Major segment area = 71.375 cm² (≈ 71.38 cm²).