Question

Find the area of shaded region, if diameters AO, OB, AB are 28 cm, 14 cm, 42 cm respectively.

• A. 693 $c{m}^2$
• B. 770 $c{m}^2$
• C. 1001 $c{m}^2$
• D. 1078 $c{m}^2$

Answer 1

The correct option is D 1078 $c{m}^2$

Diameters AO, OB, AB are 28, 14, 42 respectively in cm.
Hence corresponding radii will be 14, 7, 21 in cm
( as radius = $\frac{diameter}{2}$)
Area of shaded region
= Area of semicircle with radius 21 cm
+ Area of semicircle with radius 14 cm
+ Area of semicircle with radius 7 cm

We know,
Area of semicircle = $\frac{1}{2}$$\times$pi$\times$$r^2$

Area of semicircle with radius 21 cm
=$\frac{1}{2}$$\times$pi$\times$$r^2$= $\frac{1}{2}$$\times$$\frac{22}{7}$$\times$${21}^2$= $\frac{22\times {21}^2}{2\times 7}$= 693 $c{m}^2$

Area of semicircle with radius 14 cm
=$\frac{1}{2}$$\times$pi$\times$$r^2$= $\frac{1}{2}$$\times$$\frac{22}{7}$$\times$${14}^2$= $\frac{22\times {14}^2}{2\times 7}$= 308 $c{m}^2$

Area of semicircle with radius 7 cm
=$\frac{1}{2}$$\times$pi$\times$$r^2$= $\frac{1}{2}$$\times$$\frac{22}{7}$$\times$$7^2$= $\frac{22\times 7^2}{2\times 7}$= 77 $c{m}^2$

Area of shaded region = 693+308+77
= 1078 $c{m}^2$