Find the area of an equilateral triangle inscribed in a circle with a circumference of 18 Pi cm.
+2666 Recall that a circle's radius can be found with the circumference using the formula \(2 \pi r = 18 \pi\).
Solving for r, we find the circle's radius is 9.
Now, remember that a triangle's circumradius is defined by the formula \(abc \over 4 \times \text{Area}\).
However, we have an equilateral triangle, so we have: \(\large{s^3 \over { 4 \times {\sqrt 3 \over 4}s^2}} = 9\).
This equation simplifies to \({s \over \sqrt 3 } = 9\), meaning the side length of the triangle is \(9 \sqrt 3 \)
Now, using the formula for the area of an equilateral triangle, we have: \((9 \sqrt3)^2 \times {\sqrt 3 \over 4} = \color{brown}\boxed{243 \sqrt 3 \over 4}\)
