# How doy ou find the area of an equilateral triangle inscribed in a circle with a circumference of 64(pi) cm?

Jun 1, 2015

If the circumference of a circle is $64 \pi$ cm. then it's diameter is $64$ cm. (since circumference $= 2 \cdot$diameter).

Based on the diagram (above).

The sides of this equilateral triangle have a length of
$s = \cos \left({30}^{\circ}\right) \cdot 64$
$\textcolor{w h i t e}{\text{XXXXX}}$and since $\cos \left({30}^{\circ}\right) = \frac{\sqrt{3}}{2}$
$= 32 \sqrt{3}$

The height of the equilateral triangle is
$h = \cos \left({30}^{\circ}\right) \cdot s$

$= \frac{\sqrt{3}}{2} \cdot 32 \sqrt{3}$

$= 48$

The area of the equilateral triangle is
$A = \frac{1}{2} s h$

$= \frac{1}{2} \cdot 32 \sqrt{3} \cdot 48$

$= 768 \sqrt{3}$