each leg of an octagon is 48" what is the diameter?

Answer 1

Formula for long diameter (vertex to opposite vertex),
d = (2 * side length * sin 67.5)/(sin 45)

Formula for short diameter (one side to opposite side)
d = ((2 * side length)/(sqrt(2))) + side length

I'll do short diameter,
d = (2 * 48)/(sqrt2)) + 48
d = (96)/(1.4142) + 48
d = 67.8829 + 48
d = 115.8829

I'm not sure if this is the way you were taught.

You can make use of the fact that if you draw a short diameter in the octagon, two 45-45-90 triangles are formed (with a rectangle between them).

The short diameter would then be, x + 48 + x.

x = one of the equal legs of the 45-45-90 triangle. The side length of 48 is the hypotenuse of each of the right triangles.

Using the properties of the sides being in the proportion of x : x : x(sqrt(2)),
you can find x (since you know the hypotenuse is 48).

To find the long diameter, another right triangle is formed.

The 'short diameter' found above is one leg, 48 is the other leg and the hypotenuse is the 'long diameter'.

Since you know two legs, you can use the Pythagorean theorem to find the long diameter.