In Fig., 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 OA = 7 cm, find the area of the shaded region.

We know that,  formula for the area of the sector of a circle

Area of the sector = $\frac{\theta }{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

Area of circle = ${\mathrm{\pi r}}^2$

In the given figure we observe that,

AB and CD are diameters of the the circle

Thus, OA = OB = OD = OC = 7 cm = Radius of the circle = r

and AB = 2r = 14 cm

We get, the radius of the shaded region = R = OD/2 = 7/2 cm

Hence, the area of shaded circular region = ${\mathrm{\pi r}}^2$

= 22/7 $\times$ 7/2 $\times$ 7/2

= 77/2 

= 38.5 cm²

Now we find the area of the shaded segment of larger region = Area of semicircle $-$ Area of triangle ABC

= 1/2 $\times$ $\mathrm{\pi }$ (OA)² $-$ 1/2 $\times$ AB $\times$OC

= 1/2 $\times$ $\mathrm{\pi }$ (r)² $-$ 1/2 $\times$ 2r $\times$r

= 1/2 $\times$ 22/7 $\times$  (7)² $-$ 1/2 $\times$ 14 $\times$7

= 77 $-$ 49

= 28 cm²

Therefore, We get the formula for area of shaded region as,

Area of shaded region = Area of circular region $+$ Area of the shaded segment of larger region 

= 38.5 $+$ 28

= 66.5 cm²