Volume of a Torus Formula

A torus is a donut-shaped, 3D surface generated by rotating a circle around an axis outside the circle. The volume of a torus depends on two measurements: the major radius, which is the distance from the center of the tube to the center of the torus, and the minor radius, which is the radius of the tube itself.

Here’s the formula for calculating the volume of a torus, along with detailed explanations and examples.

Formula for the Volume of a Torus

If the major radius (distance from the center of the torus to the center of the tube) is \( R \) and the minor radius (radius of the tube) is \( r \),

The volume \( V \) of the torus can be calculated with the formula:

\( V = 2 \pi^2 R r^2 \)

In this formula:

  • \( R \) is the major radius, or the distance from the center of the torus to the center of the tube
  • \( r \) is the minor radius, or the radius of the circular tube
  • \( \pi \) (Pi) is approximately equal to 3.14159

Detailed Explanation of the Formula

The formula \( V = 2 \pi^2 R r^2 \) calculates the volume of a torus by considering the circular cross-section of the tube and the distance it travels around the center of the torus. Specifically:

  • The cross-sectional area of the tube is \( \pi r^2 \), as it is a circle with radius \( r \).
  • The circumference of the path traveled by the center of the tube is \( 2 \pi R \), where \( R \) is the major radius.
  • The total volume is then obtained by multiplying the cross-sectional area by the distance traveled, giving \( V = 2 \pi^2 R r^2 \).

Example: Calculating the Volume of a Torus

Problem: Find the volume of a torus with a major radius \( R = 10 \, \text{cm} \) and a minor radius \( r = 3 \, \text{cm} \).

Solution:

  1. Write down the formula: \( V = 2 \pi^2 R r^2 \).
  2. Substitute \( R = 10 \) and \( r = 3 \): \( V = 2 \pi^2 \times 10 \times 3^2 \).
  3. Calculate \( r^2 \): \( V = 2 \pi^2 \times 10 \times 9 \).
  4. Multiply: \( V = 180 \pi^2 \).
  5. Approximate \( \pi^2 \approx 9.8696 \): \( V \approx 180 \times 9.8696 = 1776.53 \, \text{cm}^3 \).

The volume of the torus is approximately \( 1776.53 \, \text{cm}^3 \).

This example demonstrates how to calculate the volume of a torus using the formula \( V = 2 \pi^2 R r^2 \), with given values for the major and minor radii.