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.

Question

Violet Aldrich · Accepted Answer

Perimeter #=36.33# cm.

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

#A_("circle") = pi*r^2color(white)("XXX")rarrcolor(white)("XXX")r=sqrt(A/pi)#

We are told
#color(white)("XXX")A=152 "cm"^2#
and to use
#color(white)("XXX")pi = 22/7#

#rArr r= 7# (after some minor arithmetic)

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

#color(white)("XXX")t=r*cos(60^@)#

#color(white)("XXXx")=7*sqrt(3)/2#

and
#color(white)("XXX")s=2t = 7*sqrt(3)#

#color(white)("XXXx")=12.11# (since we are told to use #sqrt(3)=1.73#)

Perimeter #=3s#

#color(white)("XXXXXX")=3 xx 12.11 = 36.33#

Aaliyah Anderson · Answer

undefined The radius $ r $ of the inscribed circle can be found using the formula $ A = \frac{\sqrt{3}}{4} \times s^2 $, where $ A $ is the area of the equilateral triangle and $ s $ is the length of its side. Then, the perimeter $ P $ of the triangle can be calculated as $ P = 3s $.

Given that $ A = 154 $ square centimeters, and $ A = \pi r^2 $, solve for $ r $. Then, using $ r $, find the side length $ s $ of the equilateral triangle, and finally compute the perimeter $ P $.

$$ r = \sqrt{\frac{A}{\pi}} $$
$$ s = \frac{2r}{\sqrt{3}} $$
$$ P = 3s $$

Substitute the given values to find $ r $, then use $ r $ to find $ s $, and finally calculate $ P $.

$$ r = \sqrt{\frac{154}{\frac{22}{7}}} $$
$$ s = \frac{2 \times \sqrt{\frac{154}{\frac{22}{7}}}}{1.73} $$
$$ P = 3 \times \frac{2 \times \sqrt{\frac{154}{\frac{22}{7}}}}{1.73} $$

Evaluate these expressions to find the perimeter $ P $.