Question

In the given figure, square ABCD is inscribed in the sector A - PCQ. The radius of sector C - BXD is 20 cm. Complete the following activity to find the area of shaded region

Answer 1

Side of square ABCD = radius of sector C-BXD = $\boxed{20}$ cm

Area of square = (side)² = ${\boxed{20}}^2$ = $\boxed{400}$ cm²

Area of shaded region inside the square

= Area of square ABCD − Area of sector C-BXD

= $\boxed{400}$ $-\frac{\theta }{360°}\times \mathrm{\pi }{r}^2$

= $\boxed{400}$ $-\frac{90°}{360°}\times \frac{3.14}{1}\times \frac{400}{1}$

= $\boxed{400}$ − 314

= $\boxed{86}$ cm²

Radius of bigger sector = Length of diagonal of square ABCD = $20\sqrt{2}$ cm

Area of the shaded regions outside the square

= Area of sector A-PCQ − Area of square ABCD

= A(A-PCQ) − A($\boxed{}$ ABCD)

= $\frac{\theta }{360°}\times \mathrm{\pi }\times r^2$ − ${\boxed{20}}^2$

= $\frac{90°}{360°}\times 3.14\times {\left(20\sqrt{2}\right)}^2$ − ${\left(20\right)}^2$

= $\boxed{628}$ − $\boxed{400}$

= $\boxed{228}$ cm²

∴ Total surface area of the shaded region = 86 + 228 = 314 cm²