Question

In given figure the area of an equilateral triangle $ABC$ is $17320.5c{m}^2$. With each vertex of the triangle as a center, a circle is drawn with a radius equal to half the length of the side of the triangle. Find the area of the shaded region. (Use $\pi =3.14$ and $\sqrt{3}=1.73205$)

Answer 1

Given $AB=BC=AC$

Area of Equilateral $△$ $ABC$ $=$ $17320.5$$c{m}^2$

$\therefore$ $\frac{\sqrt{3}}{4}\times A{B}^2$ $=17320.5$

$\therefore$ $AB$ $=200cm$

Also, $AB=2AD$

$\therefore$ $AD=100$ $cm$ $=radius$
Area of sector $DAE$ $+$ Area of sector $DBF$ $+$ Area of sector $FCE$

We know that area of sector $=\frac{\theta }{360}\times \pi \times r^2$

$=3\times \frac{60}{360}\times 3.14\times 100\times 100$

$=15700c{m}^2$
$\therefore$ Area of the shaded region $=$ Area of equilateral triangle $-$ Area of all sectors

$=17320.5-15700$

$=1620.5c{m}^2$