How do you measure arc length?

Answer 1

A curve in the plane can be measured by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment (using Pythagorean Theorem for example), the total length of the approximation can be found by summing the lengths of each linear segment.

If the curve is not already a polygonal path, better approximations to the curve can be obtained by following the shape of the curve increasingly more closely. The approach is to use an increasingly larger number of segments of smaller lengths. The lengths of the successive approximations do not decrease and will eventually keep increasing---possibly indefinitely, but for smooth curves this will tend to a limit as the lengths of the segments get arbitrarily small.

For some curves there is a smallest number L that is an upper bound on the length of any polygonal approximation. If such a number exists, then the curve is said to be rectifiable and the curve is defined to have arc length L.

For a circular arc the angle distended (in radians) times the radius is the length of the arc.