3305.52 – Stop Sign

Each of the eight sides of this stop sign is 12.5 inches long. How tall is the sign from top to bottom? (It's a regular octagon.)

Solution

If a side is $x$ , the height of the sign is

$\begin{align*}x + \dfrac{2x}{\sqrt{2}} &= x + \sqrt{2} x \\&= x(1 + \sqrt{2})\end{align*}$

A side of $12.5$ inches gives a height of $12.5(1 + \sqrt{2}),$ or about $30.2$ inches.

A variant problem, perhaps for a quiz, might be: given the height, how long is a side?