Each side of the octagon is approximately in?

Draw a couple of the triangles that make up this octagon.  To do this, draw radii from the center of the figure to the vertices.  The hypotenuse of each such triangle is 14"/2, or 7".  The central angle of each triangle is 45 deg.

See if you can verify the following:  sin (45 deg / 2) = opp / hyp = (x/2)/7.
Solve this relationship for x, which represents the length of each of the eight sides of the octagon:

sin 22.5 deg = (x/2)/7, or    sin 22.5 deg = x/14, or x = 14sin 22.5 deg (in inches).  Round off your answer to the nearest 1/100 th of an inch.