The altitude of an equilateral triangle measures 6√3 cm. What is the perimeter of the triangle?

An altitude of an equilateral triangle splits it into two 30°–60°–90° triangles whose hypotenuse is the side $s$ and whose short leg is $s/2$. The altitude is the side opposite the 60° angle, so it equals (√3/2)$s$. Setting (√3/2)$s=6$√3 gives $s=12$ cm, so the perimeter is $3\times 12=36$ cm. Treating the altitude as a side would give a perimeter of 18√3 instead.