Torus Volume Calculator Formula

Understand the math behind the torus volume calculator. Each variable explained with a worked example.

Use the Torus Volume Calculator

Formulas Used

Volume

volume = 2 * pow(pi, 2) * big_r * pow(small_r, 2)

Surface Area

surface_area = 4 * pow(pi, 2) * big_r * small_r

Variables

VariableDescriptionDefault
big_rMajor Radius (R)5
small_rMinor Radius (r)2

How It Works

Torus Formulas

Volume

V = 2 × pi² × R × r²

Surface Area

SA = 4 × pi² × R × r

Where:

  • R = major radius (distance from the center of the torus to the center of the tube)
  • r = minor radius (radius of the tube itself)

    • A torus is the 3D shape formed by revolving a circle around an axis in the same plane.

    Worked Example

    Torus with major radius 5 and minor radius 2.

    big_r = 5small_r = 2
    1. 01Volume = 2 × pi² × 5 × 4 = 40pi² ≈ 394.7842
    2. 02Surface area = 4 × pi² × 5 × 2 = 40pi² ≈ 394.7842

    Frequently Asked Questions

    What is a torus?

    A torus is a doughnut-shaped surface of revolution. It is generated by rotating a circle (with radius r) around an axis at distance R from the center of the circle.

    Can the volume and surface area be equal?

    Yes, when R × r = R, which simplifies to r = 1 (in the appropriate units). The example above coincidentally has equal numerical values.

    What is R and r in a torus?

    R (major radius) is the distance from the center of the torus to the center of the tube. r (minor radius) is the radius of the tube. R must be greater than r for a ring torus.

    Ready to run the numbers?

    Open Torus Volume Calculator