Torus surface area and volume are calculated by the Pappus's centroid theorems:

, where s - the arc length of the curve being rotated, d is the distance traveled by the centroid of the region in one rotation,

, where A - area of the region being rotated, d is the distance traveled by the centroid of the region in one rotation,

So surface area of torus:

Torus volume:

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The three different classes of standard tori correspond to the three possible relative sizes of r and R. When R > r, the surface will be the standard ring torus. The case R = r corresponds to the horn torus, which in effect is a torus with no "hole". The case R < r describes a self-intersecting surface called a spindle torus.

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