Draw the octagon, all sides of which are 10 inches.
I'll just indicate that the bottom side is 10:
 

Now temporarily, connect the vertices
to the center:

I did that just to show that each
of those 8 angles at the center are
° = °, so that if we
erase all but the bottom two, like this:



Now we know that the angle in the
above is °


Now draw in an apothem, the line from
the center to the midpoint of the bottom
 side, and label it .


Since the sides
of the octagon are 10 each, the two parts of
the bottom side are 5 each. Also the 45° angle
is bisected into two angles which are 22.5° each




So lets take away everything but
just this little right triangle:



Then we just do a little trig on that triangle:

The side opposite the 22.5° angle is 5 and the
side adjacent to it is a, so



Multiply both sides by a:



Divide both sides by tan(22.5°):





or, to the nearest tenth,

 inches.

Edwin