The regular octagon is made up of 8 iscoseles triangles, the equal sides of which are equal to the radius 10 and hence you can work out the length of the other side from this.

The angle at the center is 360/8 =45

and the angles at the base of each triangle is (180-45)/2=67.5

so if you use the sine rule you get

x/sin45=10/sin67.5

and hence x = 8.51