If the perimeter is 9 inches, each side is 3 inches.
The height of an equilateral triangle splits the triangle in two right sub-triangles: for example, consider a triangle ABC and the height AH starting from A, perpendicular to BC.
The triangle AHC is a right triangle, whose legs are AH and HC, and whose hypotenuse is AC.
AH is half the side, and AC is a side. So, we can use the Pythagorean theorem to find the height:
[tex]AH = \sqrt{AC^2-AH^2}=\sqrt{l^2-\dfrac{l^2}{4}}=\dfrac{\sqrt{3}l}{2}[/tex]
So, the area is
[tex]\dfrac{AH\cdot HC}{2} = \dfrac{l\cdot\frac{\sqrt{3}l}{2}}{2} = \dfrac{l^2\sqrt{3}}{4} = \dfrac{9\sqrt{3}}{4}[/tex]
