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is anyone knows how to calculate the area of a ring that is twisted, for example an edge of a tooth of a thread coil, please see a picture (area between A and B points)

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If you're looking for the area of a right circular conical frustum, the lateral surface area is equal to

$$\pi(d+D)\sqrt{(D-d)^2+h^2}$$

where $d,D$ are the two diameters and $h$ the height. See Wikipedia article for more details.