Question

In the given figure, ABCD is a rectangle inscribed in a circle having length 8 cm and breadth 6 cm. If π = 3.14, then the area of the shaded region is


(a) 264 cm²
(b) 266 cm²
(c) 272 cm²
(d) 254 cm²

Answer 1

All options are incorrect; the correct answer is 30.5 cm.

Join AC.
Now, AC is the diameter of the circle.
$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ A{C}^2=A{B}^2+B{C}^2\left[\mathrm{By}\mathrm{Pythagoras}\text{'}\mathrm{theorem}\right] \\ \Rightarrow A{C}^2=\left\{{\left(8\right)}^2+{\left(6\right)}^2\right\}{\mathrm{cm}}^2 \\ \Rightarrow A{C}^2=\left(64+36\right){\mathrm{cm}}^2 \\ \Rightarrow A{C}^2=100{\mathrm{cm}}^2 \\ \Rightarrow AC=10\mathrm{cm} \end{gathered}$$
∴ Radius of the circle$=\frac{10}{2}\mathrm{cm}$
=5 cm
Now,
Area of the shaded region = Area of the circle with radius 5 cm $-$ Area of the rectangle ABCD
$$\begin{gathered} =\left|\left(3.14\times 5\times 5\right)-\left(8\times 6\right)\right|{\mathrm{cm}}^2 \\ =\left|\left(\frac{314}{100}\times 25\right)-48\right|{\mathrm{cm}}^2 \\ =\left(\frac{157}{2}-48\right){\mathrm{cm}}^2 \\ =\frac{61}{2}{\mathrm{cm}}^2 \\ =30.5{\mathrm{cm}}^2 \end{gathered}$$