An octagonal stop sign can be cut from a square sheet of metal, 750mm on a side. To the nearest hundredth, what percentage of the metal square is wasted when the octagon is of maximum size ?

Answer 1

Clearly the maximum area is when four of the sides of the octagon are along the sides of the square.

Thus, that leaves us with four unused triangles, each of side s where

s/√2 + s + s/√2 = 750
s = 750/(1+√2)

The area of each such triangle is thus
(750/(1+√2))^2/2, with a total area of
2(750/(1+√2))^2

The ratio of unused area to total area is thus

2(750/(1+√2))^2
--------------------
750^2
= 6-4√2
= 34.3%

Hmmm. Seems kind of high to me. Better double-check my math.