How does one find the length of sides of an octagon given its diameter?

Given: ABCDEFGH is a regular octagon; diameter (diagonal) = d

Let the side length of the octagon be s.

Label with O the intersection of the diagonals.

Look at the formed isosceles triangle AOB.

Draw the apothem OK, which is also the altitude of the triangle AOB, which bisect the side AB.

Look at the right triangle AOK. Here we have:

The length of AK is s/2.

The length of AO is d/2.

The angle A is 67.5° (135°/2)

So that,

cos 67.5° = (s/2)/(d/2)

.383 = s/d

(.383)d = (s/d)(d)

(.383)d = s

Thus, the side length of the octagon equals (.383)d.