Question

Two circles are drawn inside a big circle with diameters $\frac{2}{3}\mathrm{rd}$ and $\frac{1}{3}\mathrm{rd}$ of the diameter of the big circle as shown in Fig. 18. Find the area of the shaded portion, if the length of the diameter of the circle is 18 cm.

Answer 1

Let the left circle be denoted as the 1st circle and the right circle be denoted as the 2nd circle.
Diameter of the big circle = 18 cm
Radius of the big circle = 9 cm
Diameter of the 1st circle = $\frac{2}{3}\times 18=12\mathrm{cm}$
Radius of the 1st circle = 6 cm
Diameter of the 2nd circle = $\frac{1}{3}\times 18=6\mathrm{cm}$
Radius of the 2nd circle = 3 cm
Area of the 1st circle = $\mathrm{\pi }(6{)}^2=36{\mathrm{\pi cm}}^2$
Area of the 2nd circle = $\mathrm{\pi }(3{)}^2=\mathrm{\pi }\times 3\times 3=9\mathrm{\pi }{\mathrm{cm}}^2$
Area of the big circle = $\mathrm{\pi }(9{)}^2=\mathrm{\pi }\times 9\times 9=81\mathrm{\pi }{\mathrm{cm}}^2$
Area of the shaded portion = Area of the big circle - (Area of the I^st circle + Area of the II^nd circle)
Area of the shaded portion = $81\mathrm{\pi }-(36\mathrm{\pi }+9\mathrm{\pi })=36\mathrm{\pi }$ cm².