In Fig. 12.27, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If `O A=7\ c m`, find the area of the shaded region

Explanation

We use the concepts of areas of circles and areas of triangles to solve the problem.
AB and CD are diameters of the circle with center O
∴ OD = OC = OA = OB =
Radius of the circle R = 7 cm
∴ AB = 2R = 14 cm
Radius of shaded circular region, r = OD/2 = 7/2 cm
Area of the shaded smaller circular region = `pi r^@`
= `pi (7/2 cm)^2`
= `22/7 × 7/2 × 7/2 cm^2`= `77/2 cm^2`= `38.5 cm^2`
Area of the shaded segment of larger circular region = Area of semicircle ACB - Area of ΔABC
= `1/2 pi(OA)^2 - 1/2 × AB × OC`= `1/2 pi R^2 - 1/2 × 2R × R`
= `1/2 × 22/7 × 72 - 1/2 × 14 × 7`
= `77 - 49`
= `28 cm^2`
Area of the shaded region = Area of the shaded smaller circular region + Area of the shaded segment of larger circular region
= `38.5 cm^2 + 28 cm^2`
= `66.5 cm^2`