Torus Volume Calculator

Calculate the volume of a torus from its major radius R and minor radius r with detailed step-by-step computation.

Enter Torus Dimensions

Result

Volume
--
cubic units
Surface Area--
Outer Radius (R + r)--
Inner Radius (R - r)--
Cross-Section Area--

Step-by-Step Computation

V = 2π²Rr²

How to Calculate Torus Volume

The volume of a torus can be derived using Pappus' centroid theorem: the volume of a solid of revolution is equal to the cross-sectional area multiplied by the distance traveled by the centroid. For a torus, the cross-section is a circle of radius r and the centroid travels a distance of 2πR.

Derivation

Step 1: Cross-Section Area

The cross-section of a torus tube is a circle with radius r.

A = πr²

Step 2: Path Length

The centroid of the cross-section travels around a circle of radius R.

Path = 2πR

Step 3: Apply Pappus' Theorem

Volume equals area times the distance traveled by the centroid.

V = A x Path = πr² x 2πR = 2π²Rr²

Relationship Between Surface Area and Volume

For a torus with major radius R and minor radius r:

  • Volume: V = 2π²Rr²
  • Surface Area: SA = 4π²Rr
  • The ratio SA/V = 2/r, which depends only on the tube radius.

Real-World Applications

Torus volume calculations are essential for designing O-rings and gaskets (calculating material volume), inflatable structures like inner tubes and life preservers, tokamak plasma containment vessels in fusion research, and toroidal tanks used in aerospace engineering.