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Torus

Volume and surface area of torus.

Torus
Torus

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

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

A = s d, 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,
V = A d, 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: S = (2\pi r)(2\pi R) = 4 \pi^2 Rr
Torus volume: V = (\pi r^2)(2\pi R)=2 \pi^2 R r^2

PLANETCALC, Torus

Torus

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

Digits after the decimal point: 5
Surface area
 
Volume
 
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Standard torus to horn torus and spindle torus transformation
Standard torus to horn torus and spindle torus transformation

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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