Arclength surprise - mathematics, Mathematics

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Suppose a unit circle, and any arc S on the unit circle in the first quadrant. No matter where S is provided, the area between S and the x-axis plus the covered area between S and y-axis is constant! Moreover, that constant is same to the length of S:

A + B = s₂ - s₁.

In Figure, note that regions A and B overlap; in that part the area is counted double times. The quantity (s₂ - s₁) shows the length of S along the arc from s₂ to s₁.

1605_arc length.gif

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