Torus Volume Calculator

The volume of a torus is V = 2π²Rr², where R is the major radius (center of torus to center of tube) and r is the minor radius (tube radius).

Pappus's theorem states that the volume of a solid of revolution equals the generating area (πr² for a circle) times the distance traveled by its centroid (2πR).

A cylinder has V = πr²h. A torus has V = 2π²Rr². The torus can be thought of as a cylinder bent into a circle, where h = 2πR.

Volume V = 2π²Rr² and surface area SA = 4π²Rr. The ratio V/SA = r/2, meaning volume grows faster than surface area as r increases.

No, different combinations of R and r can give the same surface area but different volumes. You need both radii to determine volume uniquely.

Applications include calculating capacity of toroidal tanks, material volume in O-rings and seals, plasma volume in tokamak reactors, and food production (donuts).

This calculator uses high-precision floating-point arithmetic and provides results accurate to multiple decimal places for any valid R and r values.

As R — 0, the torus degenerates into a sphere of radius r. The volume approaches (4/3)πr³.

A horn torus has R = r. Its volume is V = 2π²r³. It is the limiting case between a ring torus (R > r) and a spindle torus (R < r).

To convert between cubic units, multiply by the appropriate conversion factor. For example, cm³ to m³: multiply by 10⁻⁶.

The centroid (center of mass) of a uniform torus is at the geometric center, which is the center of the hole.

Volume scales linearly with R and quadratically with r. Doubling R doubles the volume; doubling r quadruples it.

For a toroidal shell, compute the difference between the outer torus volume and inner torus volume using their respective minor radii.

The cross-sectional area is πr², where r is the minor radius. The volume equals this area times the circumference of the major circle (2πR).

A spindle torus has R < r, creating a self-intersecting shape. The standard volume formula V = 2π²Rr² still applies to the outer surface.

A sphere of radius r has V = (4/3)πr³. A torus with R = r has V = 2π²r³ ≈ 19.74r³, which is about 4.7 times the sphere volume.

V = C_major × C_minor × r / (4π), where C_major = 2πR and C_minor = 2πr are the two circumferences.

No, this calculator is specifically for tori (donut shapes). For oblate spheroids, use an ellipsoid volume calculator.

In toroidal coordinates (ρ, φ, θ), the volume element is dV = ρ(R + ρ cos θ) dρ dφ dθ, integrating over the tube radius and angles.

The Torus Volume Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Torus Volume Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.