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# geometry question

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Find the area of an equilateral triangle inscribed in a circle with a circumference of 18 Pi cm.

Jun 28, 2022

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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: $$(9 \sqrt3)^2 \times {\sqrt 3 \over 4} = \color{brown}\boxed{243 \sqrt 3 \over 4}$$

Jun 28, 2022