Find the area of an equilateral triangle (regular 3-gon) with the given measurement. 3-inch radius a = sq. in.

So for you can solving this exercise you need to know that the radius of the circle inscribed around of an equilateral triangle is equal with 2/3 part of the height (apothem) of this equailateral triangle 

using this knowledge you can calculi easy the height of this equailateral triangle

so will be 

3 in   mean 2/3 part 

x mean       1/3 part 


x = 3*(1/3) /(2/3) = 1 /(2/3) = 3/2 

so in this case result that the length of apothem or the length of the height of this equilateral triangle will be equal (2/3)+(3/2) = (4+9)/6 = 13/6

bc. the opposite angle to apothem has measure of 60 degrees so we can writing   

sin60 = apothem divide side length 

sqrt3 /2 = (13/6) /side length 

side length *(sqrt3 /2) = 13/6

side length = (13/6) / (sqrt3 /2)

side length = (13/6) *(2/sqrt3)

side length = (13/3)*(1/sqrt3) 

side length = (13/3)*(sqrt3 /3)

side length = (13sqrt3)/9

so area = ((13sqrt3)/9 *13/6) /2

area = ((169sqrt3)/54)/2

area = (169sqrt3)/27

The answer is