Find the area of the shaded portion of an equilateral triangle with sides 6 units shown in the following figure. A circle of radius 1 unit is centred at midpoint of a side of the triangle.

Solution (By Examveda Team)

$$\eqalign{ & {\text{Area of half }}\Delta = \frac{1}{2}\left( {\frac{{\sqrt 3 }}{4}{a^2}} \right) \cr & = \frac{1}{2} \times \frac{{\sqrt 3 }}{4} \times 6 \times 6 \cr & = \frac{{9\sqrt 3 }}{2} \cr & {\text{Area of shaded region}} \cr & \Rightarrow \frac{{9\sqrt 3 }}{2} - \frac{{90}}{{360}}\pi {r^2} \cr & = \frac{{9\sqrt 3 }}{2} - \frac{1}{4} \times \frac{{22}}{7} \times 1 \times 1 \cr & = \frac{{9\sqrt 3 }}{2} - \frac{{11}}{{2 \times 7}} \cr & = \frac{1}{2}\left( {9\sqrt 3 - \frac{{11}}{7}} \right){\text{c}}{{\text{m}}^2} \cr} $$