Torus

A torus has the shape of a doughnut. A torus is a surface of revolution generated by revolving a circle in three-dimensional space about a line that does not intersect the circle.

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

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

So surface area of torus:
Torus volume:

Torus

Radius of the tube (r)

Radius of torus

Distance from the center of the tube to the center of the torus

Calculation precision

Digits after the decimal point: 5

Calculate

Surface area

 

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 is a torus with no "hole". The case R < r describes a self-intersecting surface called a spindle torus.