Question

In the given figure, a circle of diameter 21 cm is given. Inside this circle, two circles with diameters $\frac{2}{3}$ and $\frac{1}{3}$ of the diameter of the big circle have been drawn, as shown in the given figure. Find the area of the shaded region.

Answer 1

Diameter of the big circle = 21 cm
Radius = $\left(\frac{21}{2}\right)$ cm = 10.5 cm
∴ Area of the bigger circle = ${\mathrm{\pi r}}^2$ = $\left(\frac{22}{7}\times 10.5\times 10.5\right)$ cm²
= 346.5 cm²


Diameter of circle I = $\frac{2}{3}$ of the diameter of the bigger circle
= $\frac{2}{3}$ of 21 cm = $\left(\frac{2}{3}\times 21\right)$ cm = 14 cm
Radius of circle I (r₁) = $\left(\frac{14}{2}\right)$ cm = 7 cm
∴ Area of circle I = ${{\mathrm{\pi r}}_1}^2$ = $\left(\frac{22}{7}\times 7\times 7\right)$ cm²
= 154 cm²

Diameter of circle II = $\frac{1}{3}$ of the diameter of the bigger circle
= $\frac{1}{3}$ of 21 cm = $\left(\frac{1}{3}\times 21\right)$ cm = 7 cm
Radius of circle II (r₂) = $\left(\frac{7}{2}\right)$ cm = 3.5 cm
∴ Area of circle II = ${{\mathrm{\pi r}}_2}^2$ = $\left(\frac{22}{7}\times 3.5\times 3.5\right)$ cm²
= 38.5 cm²

∴ Area of the shaded portion = {Area of the bigger circle - (Sum of the areas of circle I and II)}
= {346.5 - (154 + 38.5)} cm²
= {346.5 - 192.5} cm²
= 154 cm²
Hence, the area of the shaded portion is 154 cm²