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A circle contains 360 degrees. Draw a circle, add a radius to any point on it, and then rotate the radius completely around. After it has returned to the point it initially intersected the curve, the radius will have rotated through 360 degrees.

(another explanation, maybe no better.)

A complete circle measures 360 degrees, so a half-circle is 180 degrees, a quarter-circle is 90 degrees and so forth. Another way to look at it is in terms of the central angle formed by the radii drawn from each endpoint of the arc - the measure of the arc in degrees is the same as the measure of this central angle in degrees.

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