They would be 22.5* ,but just guessing!!
360/8= 45 so that would be divided by 2= 22.5

Approximately 4' long, again just guessing

Chap- Just saw your post. Jake can answer it better than I. I will explain the correct angles that Jake provided. 8 sides will divide up a circle into 45 degree segments. But, that angle is divided for each half to miter together...as Jakes 22.5 degrees show.

Place a timber in a miter box and cut 22.5 degree angles on each end. You can play with the length for the desired size of the fire pit.

Chap- If you want exactly 10 feet across two opposite sides....outside of 4x4's....the long point of each side would be 49 11/16 inches. That is long point to long point.

If you want 10 feet max across opposite corners....then each 4x4 should be cut to 45 15/16 inches. This would then measure 9' -2 7/8" across opposite sides.
Jake is very modest on his math skills. I wish I knew what he has learned!.

Damn! where were you guys when i was in algebra class? ......Oh yeah the internet had not been invented yet.

Ray- Not trying to be a smart asshole here....but we can make it larger. If you take an 8 foot 4x4...and make a 22.5 degree cut across its middle....you can effectively lengthen your 8 foot 4x4 with the overlapping ends being cut across the middle. I have stretched lumber many times this way out of necessity! The tangent of 22.5 degrees times the actual width of a 4 x4...3.5" computes to roughly 1.45". But you have to divide that in half...and allow roughly 1/8" for the saw kerf.

Basically you can get two segments out of an 8 foot 4x4 that measure 48-5/8".

Plugging this " economized stretching of this 4x 4", we now can build an octagon 9'- 9 3/8" across.