The area of the pentagon is that of a rectangle of length $b$ and breadth $a$ plus that of a triangle of base $b$ and height $(c-a)$, i.e.

$\mathrm{ab}+\frac{1}{2}b(c-a)=\frac{1}{2}b(2a+c-a)=\frac{1}{2}b(a+c)$.

[The area may also be considered to be the sum of the areas of two trapezia that have parallel sides $a$ and $c$ and whose heights have total length $b$.]