Answer
$A=\frac 52 bh$
$h=\frac{2A}{5b}$
Work Step by Step
Decompose the polygon in $5$ equal triangles with common vertex at center. Each triangle has height $h$ and base $b$, so each triangle area is:
$$A_1=\frac 12 bh$$
Area of whole polygon:
$$A=5A_1=5\cdot \frac 12 bh=\frac 52 bh$$
Solve for $h$:
$$h=A\div\left(\frac 52 b\right)=A\cdot \frac{2}{5b}=\frac{2A}{5b}$$