# Torus

Volume and surface area of torus.

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#### Timur

Created: 2008-11-28 08:28:35, Last updated: 2021-03-02 21:55:27

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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:

$A = s d$, 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: $V = (\pi r^2)(2\pi R)=2 \pi^2 R r^2$

#### Torus

Distance from the center of the tube to the center of the torus
Digits after the decimal point: 5
Surface area

Volume

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.

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