# The area of a circle inscribed in an equilateral triangle is 154 square centimeters. What is the perimeter of the triangle? Use pi=22/7 and square root of 3= 1.73.

Jan 6, 2016

Perimeter $= 36.33$ cm.

#### Explanation:

This is Geometry, so lets look at at a picture of what we are dealing with:

${A}_{\text{circle") = pi*r^2color(white)("XXX")rarrcolor(white)("XXX}} r = \sqrt{\frac{A}{\pi}}$

We are told
color(white)("XXX")A=152 "cm"^2
and to use
$\textcolor{w h i t e}{\text{XXX}} \pi = \frac{22}{7}$

$\Rightarrow r = 7$ (after some minor arithmetic)

If $s$ is the length of one side of the equilateral triangle and $t$ is half of $s$

$\textcolor{w h i t e}{\text{XXX}} t = r \cdot \cos \left({60}^{\circ}\right)$

$\textcolor{w h i t e}{\text{XXXx}} = 7 \cdot \frac{\sqrt{3}}{2}$

and
$\textcolor{w h i t e}{\text{XXX}} s = 2 t = 7 \cdot \sqrt{3}$

$\textcolor{w h i t e}{\text{XXXx}} = 12.11$ (since we are told to use $\sqrt{3} = 1.73$)

Perimeter $= 3 s$

$\textcolor{w h i t e}{\text{XXXXXX}} = 3 \times 12.11 = 36.33$