Find the area of the shaded region in Fig., where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre.

We know that, the formula for the area of the sector of a circle

Area of the sector = $\frac{\theta }{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

Where r is the radius of the circle = 6 cm

Side of the equilateral = s = 12 cm

We know each angle of an equilateral is 60⁰.

 Area of shaded region = Area of circle $+$ Area of ΔOAB $-$ Area of sector OCD

= ${\mathrm{\pi r}}^2$ $+$ $\sqrt{3}/4$ ( side $\times$ side) $-$ $\frac{\theta }{{360}^0}$ $\times$${\mathrm{\pi r}}^2$

= $\mathrm{\pi }$(6)² $+$ $\sqrt{3}/4$ (12)² - 60⁰/360⁰ $\times$ $\mathrm{\pi }$ (6)²

= 36$\mathrm{\pi }$ $+$ 36$\sqrt{3}$  - 6$\mathrm{\pi }$ 

= (30$\mathrm{\pi }$ $+$ 36$\sqrt{3}$) 

= (30 $\times$ 22/7 $+$ 36$\sqrt{3}$) 

= (36$\sqrt{3}$ $+$ 660/7) cm²