The problem asks for the perimeter of an equilateral triangle given the area of its incircle.

Incircle Area Calculation

The formula for the area of a circle is $A = \pi r^2$, where $A$ is the area and $r$ is the radius. We are given the area of the incircle as $462\text{ cm}^2$. We can use the approximation $\pi \approx \frac{22}{7}$.

$ A_{incircle} = \pi r^2 $ $ 462 = \frac{22}{7} r^2 $

Now, we solve for $r^2$: $ r^2 = 462 \times \frac{7}{22} $ $ r^2 = 21 \times 7 $ $ r^2 = 147 $

To find the radius $r$, we take the square root: $ r = \sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3}\text{ cm} $

Relating Inradius to Equilateral Triangle Side

For an equilateral triangle with side length $a$, the radius of the incircle ($r$) is related to the side length by the formula:

$ r = \frac{a}{2\sqrt{3}} $

Finding the Triangle's Side Length

Substitute the calculated value of $r$ into the formula:

$ 7\sqrt{3} = \frac{a}{2\sqrt{3}} $

Solve for $a$: $ a = 7\sqrt{3} \times 2\sqrt{3} $ $ a = 14 \times (\sqrt{3} \times \sqrt{3}) $ $ a = 14 \times 3 $ $ a = 42\text{ cm} $

Calculating the Perimeter

The perimeter ($P$) of an equilateral triangle is given by $P = 3a$. Using the calculated side length $a = 42\text{ cm}$:

$ P = 3 \times 42 $ $ P = 126\text{ cm} $

Therefore, the perimeter of the equilateral triangle is $126\text{ cm}$.