# Arc Length of Cycloid

Arc Length of Cycloid

A polygon rolls on a line . The positions of a vertex when has a side flush with form a polygonal path (orange). The orange segments are the base sides of green isosceles triangles. When you drag the "combine" slider, the green triangles combine to form a right triangle with height , the diameter of incircle of the polygon. Hence, the length of the orange path is the sum of the height and the hypotenuse of this triangle.

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2r

As the number of sides goes to infinity, the orange path approaches a half cycloid and the hypotenuse of the triangle approaches . Hence the arc length of one arc of the cycloid is .

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