Torus Calculator

What is a Torus?

A torus is a three-dimensional geometric shape that resembles a doughnut. It is characterized by two radii: the major radius (\(R\)), which is the distance from the center of the tube to the center of the torus, and the minor radius (\(r\)), which is the radius of the tube itself.

Formulas for Tori

Let \(R\) be the major radius, \(r\) be the minor radius, \(V\) be the volume, and \(A\) be the surface area. The formulas for calculating the volume and surface area of a torus are as follows:

1. Volume: \(V = ( \pi \times r^2 ) \times ( 2 \pi \times R )\)
2. Surface Area: \(A = ( 2 \pi \times r ) \times ( 2 \pi \times R )\)

Step-by-Step Calculations

Let's calculate these properties for a torus with major radius \(R = 5\) units and minor radius \(r = 2\) units:

1. Volume: \[V = ( \pi \times 2^2 ) \times ( 2 \pi \times 5 ) = ( \pi \times 4 ) \times ( 2 \pi \times 5 ) = 8 \pi^2 \times 5 = 40 \pi^2 \approx 197.92 \text{ cubic units}\]
2. Surface Area: \[A = ( 2 \pi \times 2 ) \times ( 2 \pi \times 5 ) = ( 2 \pi \times 2 ) \times ( 2 \pi \times 5 ) = 4 \pi \times 10 \pi = 40 \pi^2 \approx 789.57 \text{ square units}\]