Answer
$A=4bh$
$h=\frac{A}{4b}$
Work Step by Step
Decompose the polygon in $8$ equal triangle 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:
$$A=8A_1=8\cdot \frac 12 bh=4bh$$
Solve for $h$:
$$h=A\div(4b)=A\cdot \frac{1}{4b}=\frac{A}{4b}$$
