Question

The area of a circle inscribed in an equilateral triangle is $154c{m}^2$. Find the perimeter of the triangle. [Use $\pi =\frac{22}{7}$ and $\sqrt{3}=1.73$]

Answer 1

Let r be the radius of the inscribed circle.

Given that, Area $=154c{m}^2$

$\Rightarrow \pi r^2=154$

$\Rightarrow r=7cm$

Let a be the side and h height of the triangle. From properties of an equilateral triangle, we know that the point O divides AD in the ratio 2:1.

So, $r=\frac{h}{3}$

$\Rightarrow h=3r=21cm$

Now, altitude, $h=\frac{\sqrt{3}}{2}a$

$\Rightarrow a=\frac{2h}{\sqrt{3}}=\frac{42}{\sqrt{3}}=14\sqrt{3}cm$

Hence, perimeter $=3a=3\times 14\sqrt{3}cm=72.7cm$