Area of equilateral triangle: \(\frac{\sqrt{3}}{4}s^2\)
Area of regular hexagon: \(\frac{3\sqrt{3}}{2}s^2\)
If the perimeter of the triangle is 18, then one side is 6, therefore the area of it is,
\(\frac{\sqrt{3}}{4}\cdot36\)
\(area = 9\sqrt{3}\)
Because their areas are equal,
\(\frac{3\sqrt{3}}{2}s^2 = 9\sqrt{3}\)
Simplify,
\(3\sqrt{3}\cdot s^2 = 18\sqrt{3}\)
\(s^2 = 6\)
\(s = \sqrt{6}\)
There are 6 sides in a hexagon, so the perimeter is, \(\sqrt{6} \cdot 6 = 6\sqrt{6}\). 6 + 6 = 12, so that's the answer.
EDIT: I messed up my arithmetic
