Answer
$54.15\ unit^2$
Work Step by Step
1. Connect the middle two apexes and let its length be $x$, use the Law of Cosines, we have $x=\sqrt {(5)^2+(11)^2-2(5)(11)cos72^\circ}\approx10.58$
2. For the lower triangle, let its bottom angle be $\theta$, use the Law of Cosines, $\theta=cos^{-1}(\frac{(7)^2+(8)^2-(10.58)^2}{2(7)(8)})\approx89.5^\circ$
3. Thus the total area $A=\frac{1}{2}(5)(11sin72^\circ)+\frac{1}{2}(7)(8sin89.5^\circ)\approx54.15\ unit^2$
