Take a regular octagon with vertices labelled ABCDEFGH anticlockwise
What is the ratio of the lengths AB and AD?
The correct answer is 1 : 1 + sqrt(2), or about 1:2.414
Bad diagram:
Set the edges to length 1 for simplicity, since i only care about the ratio
Next, AD is equal to the red line plus 2 green lines
The red lines are 1, because the middle red line is just the same length as the top, which is an edge of the octagon
For the green lines use pythagorean's theorem, green^2+green^2=1^2
This gives green = 1/sqrt(2)
So AD has length 1+2/sqrt(2)
2/sqrt(2) = sqrt(2)
So AD has length 1+sqrt(2)
So the ratio is 1 : 1+sqrt(2)