Question

The area of an equilateral triangle is 17320.5 cm^2 with each vertex of triangle as a center a circle is drawn with radius equal to half the length of the side of a triangle find the area of the shaded region ( use root3=1.73205 and pi=3.14)

Answer 1

Let each side of the triangle be a cm. Then,

Area of the triangle = $\sf{ \frac{ \sqrt{3} }{4} {a}^{2} = 17320.5 {cm}^{2} }$

$\sf{{a}^{2} = \frac{17320.5 \times 4}{ \sqrt{3} } = \frac{17320.5 \times 4}{1.73205} = 40000}$

$\therefore$ $\sf{a = \sqrt{40000} cm = 200cm}$

Radius of the circle, r = $\sf{\frac{a}{2}=100cm}$

Area of the sector with r = 100cm and $\sf{\theta = 60°}$

$\sf{\frac{\theta}{360} \pi {r}^{2} = \frac{60}{360} \times 3.14 \times {(100)}^{2} = \frac{15700}{3} {cm}^{2}}$

Hence, the area of the shaded region

= ar(∆ABC) - 3 × area of the sector

= $\sf{(17320.5 - 3 \times \frac{15700}{3} ) {cm}^{2}}$

= $\sf{1620.5{cm}^{2}}$