The arc that a 15degree angle forms represents just a small fraction of a circle's circumference. The fraction between the arc length and the circle's total length exactly equals the fraction between the angle size and the degree measure of the total circle. The circumference itself is directly related to a separate length  the radius of the circle. The ratio between the circumference and the radius is twice the constant pi.

Multiply the angle's radius by 6.283, which is approximately twice pi. For this example, imagine a radius of 6 inches:
6 x 6.283 = 37.698. This is the circle's total circumference.

Multiply the circumference by 15:
37.698 x 15 = 565.47.

Divide the result by 360, the total number of degrees in the circle:
565.47 / 360 = 1.57 inches. This is the length of the angle's associated arc.
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