In an equilateral △ABC of side 14 cm, side BC is the diameter of a semi-circle as shown in the figure. Find the area of the shaded region.

Given,

Equilateral triangle side = 14 cm.

BC = 14 cm is diameter of semi-circle.

∴ Radius = $\dfrac{14}{2}$ = 7 cm.

Area of shaded region = Area of equilateral △ABC + Area of semi-circle BDC

Calculating the area of equilateral triangle ABC,

$\text{Area of equilateral triangle ABC} = \dfrac{\sqrt{3}}{4} \times \text{(side)}^2 \\[1em] = \dfrac{\sqrt{3}}{4} \times 14^2 \\[1em] = \dfrac{\sqrt{3}}{4} \times 196 \\[1em] = 1.732 \times 49 \\[1em] = 84.868 \text{ cm}^2.$

Calculating the area of semi-circle BDC,

$\text{Area of semi-circle BDC} = \dfrac{1}{2} πr^2 \\[1em] = \dfrac{1}{2} \times \dfrac{22}{7} \times 7^2 \\[1em] = \dfrac{1}{2} \times \dfrac{22}{7} \times 49 \\[1em] = 11 \times 7 \\[1em] = 77 \text{ cm}^2.$

Area of shaded region = Area of equilateral triangle ABC + Area of semi-circle BDC

= 84.868 + 77 = 161.868 cm².

Hence, area of shaded region = 161.868 cm².