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 = 20 cm

Area of square = (side)² = 20² = 400 cm² 

Area of shaded region inside the square 
= Area of square ABCD − Area of sector C-BXD
= 400  \[- \frac{\theta}{360° } \times \pi r^2\]

= 400  \[- \frac{90° }{360° } \times \frac{3 . 14}{1} \times \frac{400}{1}\]

= 400 - 314 

= 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( ABCD )

=  \[\frac{\theta}{360° } \times \pi \times r^2\] - 20² 

 = \[\frac{90°}{360°} \times 3 . 14 \times \left( 20\sqrt{2} \right)^2\] - (20)² 

= 628 - 400

= 228 cm² 

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