Answer
$h = \frac{\sqrt2}{\sqrt3}e$
Work Step by Step
To find h, we consider the lengths of each of the sides. We know that h is part of a 30-60-90 triangle, where one of the sides is e. Thus, we find that the other side is $\frac{2e}{\sqrt3}$. The other aspect of a triangle that h is a part of is a 45-45-90 triangle with sides of e, meaning that the length is $\sqrt2 e$
We use the pythagorean theorem:
$h = \sqrt{-(\frac{2e}{\sqrt3})^2 + (\sqrt2 e)^2 } = \sqrt{\frac{2e^2}{3}}=$$ \frac{\sqrt2}{\sqrt3}e$