Geometry - Torus

Torus

Definition, Properties, and Formulae of a Torus

A torus is a doughnut-shaped surface formed by revolving a circle around an axis outside the circle. It is a 3D shape with a hole in the center, characterized by two radii:

  • Major radius (R)
  • Minor radius (r)

Torus

Key Parameters

  • \( r \): Minor radius of the torus (radius of the tube)
  • \( \alpha \): Central angle of the circle used to create the torus
  • \( b \): Length of the base circle

1. Surface Area of the Torus \(A\)

\[ A = \frac{\pi r^2 \alpha}{360} = \frac{br}{2} \]

This formula calculates the surface area of the torus. It represents the outer surface formed by the revolving circle.

2. Base Circle Length \(b\)

\[ b = \frac{2 \pi r \alpha}{360} \]

This formula calculates the length of the base circle of the torus, which is essential for understanding the geometry and how the torus is formed.

Applications

  • Used in various engineering applications such as the design of tubes, pipes, and rings.
  • Common in computer graphics for creating donut shapes or 3D modeling in animation.
  • Found in advanced physics, especially in magnetic fields and toroidal reactors.
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