If the area of an incircle of an equilateral triangle is 462 $$\text{cm}^2$$​, find the perimeter of the triangle.

Correct option is D

Given:

Area of incircle = 462 cm²

Formula Used:

Let side of the equilateral triangle be a

The radius r of the incircle of an equilateral triangle is given by

r $$= \frac{a\sqrt{3}}{6}$$​

Area of a circle:

A $$= \pi r^2$$​

Solution:

​$$\pi \left( \frac{a\sqrt{3}}{6} \right)^2 = 462 \\ \ \\\pi \cdot \frac{a^2 \cdot 3}{36} = 462 \\ \ \\\frac{\pi a^2}{12} = 462 \\ \ \\a^2 = \frac{462 \times 12}{\pi} \\ \ \\a^2 = \frac{5544}{\pi} =\frac{5544}{3.1416} \approx 1764 \\ \ \\a = \sqrt{1764} = 42 \text{ cm}$$​​

Perimeter of equilateral triangle = 3a =$$3 \times 42$$​ =126 cm