My colleague's daughter wanted to be a stop sign for Halloween,
and he needed to cut out such a sign from a piece of cardboard
that was 3 feet by 4 feet. What cuts did he make so that the
stop sign (a regular octagon) was as big as possible?
Intuitively, we want to make the height as large as possible,
leaving a little left over material on the sides. (We could
alternately think of it as make the sides as large as possible
leaving a little left over for the height, it doesn't matter.)
Using this way of thinking, obviously, we're thinking of the
cardboard as having the 3-foot side running vertically and the
4-foot side running horizontally. When we see the octagon as
triangles and rectangles, we know that the interior rectangle
has the same height as the sides of the octagon, which we'll
call . Looking at the triangles, then, we want to find out what
is the side length of a right triangle with hypotenuse .
Whatever that side length is, call it , we want to double it,
add it to , and set that equal to 3. When w