Question

What is the lateral surface area of this triangular prism?

Answer 1

first off let's take a peek at what the slanted side of the base is hmmm

$\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2 \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{3}\\ b=\stackrel{opposite}{3}\\ \end{cases}\implies c^2=3^2+3^2 \\\\\\ c^2=3^2(1+1)\implies c^2=3^2(2)\implies c=\sqrt{3^2(2)}\implies c=3\sqrt{2}$

Check the picture below, the lateral surface, namely just the sides, excluding top and bottom, is really just 3 rectangles, so let's get their area and sum them up.

$\stackrel{\textit{\large Areas}}{\stackrel{two~rectangles}{2(3\cdot 6)}~~ +~~\stackrel{rectangle}{(6\cdot 3\sqrt{2})}}\implies 36+18\sqrt{2}~~ \approx ~~61.46$