Question

How do you find the area of the shaded region

Answer 1

well, first off, let's notice that we have a trapezoid with a rectangle inside it, so the rectangle is really "using up" area that the trapezoid already has.

now, if we just get the area of the trapezoid, and then the area of the rectangle alone, and then subtract that area of the rectangle, the rectangle will in effect be making a hole inside the trapezoid's area, and what's leftover, is the shaded section, that part the hole is not touching.

$\bf \textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h&=height\\ a,b&=\stackrel{parallel~sides}{bases} \\\cline{1-2} h&=8\\ a&=16\\ b&=8 \end{cases}\implies A=\cfrac{8(16+8)}{2}\implies A=96 \\\\\\ \stackrel{\textit{area of the rectangle}}{(5\cdot 3)\implies 15}~\hfill \stackrel{~\hfill \textit{shaded area}}{\stackrel{\textit{trapezoid}}{96}~~ - ~~\stackrel{\textit{rectangle}}{15}~~ = ~~81}$