Torus Calculator

Torus (Donut Shape)

A torus is a 3D surface generated by rotating a circle around an external axis in the same plane. Think of a donut, a life preserver, or an inflatable swimming ring — all are tori.

The two radii:

• R = major radius: distance from the center of the tube to the center of the torus
• r = minor radius: radius of the circular tube itself
• The minor radius must be smaller than the major radius (r < R) for a proper ring torus

Formulas (from Pappus centroid theorem):

Pappus theorem (why these formulas work): A torus is formed by rotating a circle of radius r through a full 360 degrees around an axis at distance R.

• Volume = (area of circle) × (distance traveled by centroid) = pi * r^2 × 2piR
• Surface Area = (circumference of circle) × (distance traveled) = 2pir × 2piR

Worked example — standard donut shape (R = 4 cm, r = 1.5 cm):

• Volume: 2 × pi^2 × 4 × 1.5^2 = 2 × 9.87 × 4 × 2.25 ≈ 177.7 cm^3
• Surface Area: 4 × pi^2 × 4 × 1.5 = 4 × 9.87 × 6 ≈ 236.9 cm^2

Special torus types:

• If R = r: the inner circle collapses to a point: called a “horn torus”
• If R < r: the torus intersects itself: called a “spindle torus”
• This calculator assumes R > r (the standard ring torus)

Real-world tori: Tire inner tubes, life preservers, magnetic confinement fusion reactors (tokamaks), and architectural arches all follow toroidal geometry. The torus also appears in topology — it is the surface with one hole, and it differs fundamentally from a sphere (which has none).