In the given figure, AB and CD are diameters of the circle with centre O, and are perpendicular to each other. A semi-circle with diameter AO and a circle with diameter OB are drawn inside the circle. If AO = 7 cm, then what is the area of the shaded region?

It is given that the radius of the bigger circle is 7 cm.

∴ Area of the bigger circle = π (radius)²

$=\frac{22}{7}\times 7^2=154$ cm²

Diameter of the semi-circle is AO.

⇒ Radius of the semi-circle  $=\frac{AO}{2}=\frac{7}{2} cm$

∴Area of the semi-circle = $\frac{1}{2}\mathrm{\pi }\times {\left(\frac{7}{2}\right)}^2=\frac{1}{2}\times \frac{22}{7}\times \frac{49}{4}=19.25 {\mathrm{cm}}^2$ 

Also, the diameter of inner circle is OB.

⇒ Radius of inner circle = 3.5 cm

∴ Area of inner circle = π × (radius)²

$=\frac{22}{7}\times {\left(\frac{7}{2}\right)}^2=38.5 {\mathrm{cm}}^2$

∴ Area of shaded region

= Area of bigger circle – (Area of semi-circle with diameter AO + Area of

circle with diameter OB)

= [154 – (19.25 + 38.5)] cm²

= [154 – 57.75] cm² = 96.25 cm²

Thus, the area of the shaded region is 96.25 cm² .