An equilateral triangle whose area and perimeter are
    equal has the area of exactly the square root of 432.

Let's say we have an equilateral triangle with a perimeter of 3x units. Then each side is x units, and the area is sqrt(3) * x^2 / 4 square units. So we set:

            3x = sqrt(3) * x^2 / 4
             3 = sqrt(3) * x / 4
   3*4/sqrt(3) = x
    12/sqrt(3) = x

Then the area is:

    sqrt(3) * (12/sqrt(3))^2 / 4
    sqrt(3) * 144 / 12
    sqrt(3) * 12
    sqrt(3) * sqrt(144)
    sqrt(432)

Fun!