Having a polygon inscribed in a circle divides the circle into equal parts, equal to the amount of sides the polygon has. In the case of a regular octagon inscribed in a circle, the circle is divided into 8 equal parts because of this. Since the measure of a whole circle is 360 degrees, an arc formed by an octagon is equal to 360 divided by 8.
Arc = 360 degrees / 8 = 45 degrees.