Torus Calculator
Calculate the volume and surface area of a torus (donut shape) from major and minor radii.
Uses Pappus centroid theorem formulas with a worked example.
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):
| Property | Formula |
|---|---|
| Volume | V = 2 * pi^2 * R * r^2 |
| Surface Area | A = 4 * pi^2 * R * r |
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).