Question

In Fig. 13.112, a square OABC is inscribed in a quadrant OPBQ. If OA = 15 cm, find the area of shaded region (use π = 3.14).

Answer 1

OABC is a square.

Side of the square = OA = 15 cm

We know

Length of the diagonal of square = $\sqrt{2}$ × Side of the square

∴ OB = Radius of the quadrant of the circle = Length of the diagonal of square = $15\sqrt{2}\mathrm{cm}$

Now,

Area of shaded region

= Area of quadrant OPQO − Area of the square OABC

$$\begin{gathered} =\frac{1}{4}\times \mathrm{\pi }{\left(\mathrm{OB}\right)}^2-{\left(\mathrm{OA}\right)}^2 \\ =\frac{1}{4}\times 3.14\times {\left(15\sqrt{2}\right)}^2-{\left(15\right)}^2 \\ =\frac{1}{4}\times 3.14\times 450-225 \end{gathered}$$
$$\begin{gathered} =353.25-225 \\ =128.25{\mathrm{cm}}^2 \end{gathered}$$

Thus, the area of the shaded region is 128.25 cm².