The diagonal of a square is equal to the side of an equilateral triangle. If the area of the square is $$18\sqrt{3}$$ sq. cm. What is the area (in $$cm^2$$) of the equilateral triangle?

If the area of the square is $$18\sqrt{3}$$ sq. cm.

$$side \times side = 18\sqrt{3}$$

$$\left(side\right)^2=18\sqrt{3}$$    Eq.(i)

$$(diagonal)^2 = (side)^2 +(side)^2$$

$$(diagonal)^2 = 2(side)^2$$

diagonal = $$\sqrt{ 2}\ \times side$$    Eq.(ii)

The diagonal of a square is equal to the side of an equilateral triangle.
Area of the equilateral triangle = $$\frac{\sqrt{3\ }\times\ (diagonal)^2}{4}$$

Put Eq.(ii) in the above equation.

= $$\frac{\sqrt{3\ }\times\ (\sqrt{ 2}\ \times side)^2}{4}$$

= $$\frac{\sqrt{3\ }\times\ 2(side)^2}{4}$$

Now put Eq.(i) in the above equation.

= $$\frac{\sqrt{3\ }\times\ 2\times18\sqrt{3}}{4}$$

= $$\frac{\left(36\times\ 3\right)}{4}$$

= $$9\times3$$

= 27 $$cm^2$$