The area of an equilateral triangle ABC is 17320.5 cm² . With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see Fig.). Find the area of the shaded region. (Use π = 3.14 and $\sqrt{3}$ = 1.73205)

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

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

 Area of circle = ${\mathrm{\pi r}}^2$ 

Given that,

Area of an equilateral triangle ABC = 17320.5 cm²

$\frac{\sqrt{3}}{4}$ $\times$ (side) ²=  17320.5 

(side) ² = ( 17320.5  $\times$ 4) / $\sqrt{3}$

=  ( 17320.5  $\times$ 4) / 1.73205

= $\sqrt{10000\times 4}$

= 100 $\times$2

= 200 cm

Now we find the radius

Radius = 1/2 $\times$ (side length)

=  1/2 $\times$ (200)

= 100 cm

We know that, all the interior angles of an equilateral are of 60⁰

$\theta$ =  60⁰

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

Area of the 3 sector = 3$\times$  $\frac{{60}^0}{{360}^0}$ $\times$ 3.14 $\times$ (100)²

= 15700 cm²

Area of shaded region = Area of triangle ABC $-$ Area of the 3 sectors

= 17320.5$-$-15700

= 1620.5 cm²