Calculus Tips: Arc length

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Choose a finite number of points along a curve and connect each point to the next with a straight line. The sum of the lengths of such line segments is the length of a "polygonal path".

Definition: The length of the curve is the smallest number that such lengths of polygonal paths can never exceed, no matter how close together the discretely placed endpoints of line segments are.

In the language of mathematicians, the arc length is the supremum of all lengths of such polygonal paths.

This definition does not require the curve to be "smooth"; it need not be either the graph or the image of a differentiable function.

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