Question

The area of the shaded region in the figure given alongside is:

(a) \(\frac{a^2}{2}\big(\frac{π}{2}-1\big)\) sq. units 

(b) \({a^2}\big({π}-1\big)\) sq. units 

(c) \({a^2}\big(\frac{π}{2}-1\big)\) sq. units

(d) \(\frac{a^2}{2}\big({π}-1\big)\) sq. units

Answer 1

Answer : (c) a² ( \(\frac{\pi}{2}\) - 1) sq. units. 

Since C is the center of the circle, radius of the circle = a units. 

∴ Area of the semi-circle = \(\frac{\pi a^2}{2}\) sq. units 

AC = CD = CB = radii of the circle. 

⇒ Both the triangles ΔABC and ΔBCD are isosceles and are equal.

∴ Area of each triangle = \(\frac{1}{2}\) a² 

∴ Total area of both the triangles = 2 × \(\frac{1}{2}\)a²  

= a² . sq. units. 

∴ Area of shaded region = \(\frac{\pi a^2}{2}\) - a² 

=  a² ( \(\frac{\pi}{2}\) - 1) sq. units.