We can see that if we split an equilateral triangle in half, we are left with two congruent right triangles. Thus, one of the legs of one of the right triangles is #1/2s#, and the hypotenuse is #s#. We can use the Pythagorean Theorem or the properties of #30˚-60˚-90˚# triangles to determine that the height of the triangle is #sqrt3/2s#.
If we want to determine the area of the entire triangle, we know that #A=1/2bh#. We also know that the base is #s# and the height is #sqrt3/2s#, so we can plug those in to the area equation to see the following for an equilateral triangle:
#A=1/2bh=>1/2(s)(sqrt3/2s)=(s^2sqrt3)/4#
In your case, the perimeter is #18# units. Since all #3# sides of the triangle are congruent, we can say that #s=6#, since #3s=18#.
#A=(6^2sqrt3)/4=(36sqrt3)/4=9sqrt3# #"units"^2#
