Torus Surface Area Calculator

The surface area of a torus is SA = 4π²Rr, where R is the major radius (center of torus to center of tube) and r is the minor radius (tube radius).

The major radius R is the distance from the center of the torus to the center of the tube. The minor radius r is the radius of the tube itself.

A sphere with radius r has SA = 4πr². A torus has SA = 4π²Rr. The torus formula involves both radii and π² instead of π.

Pappus's theorem states that the surface area of a surface of revolution equals the generating curve's length times the distance traveled by its centroid: SA = (2πr)(2πR).

No, surface area and volume are independent — different combinations of R and r can give the same volume but different surface areas.

Applications include calculating material for O-rings, seals, gaskets, toroidal inductors, donut-shaped buildings, and plasma containment vessels.

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

When R = r, the torus is called a "horn torus" — the tube just touches the center. When R > r, it is a "ring torus" (normal donut). When R < r, it is a "spindle torus."

The aspect ratio is R/r. A high aspect ratio (R >> r) means a thin, elongated torus. A low aspect ratio (R ≈ r) means a fat, stubby torus.

To convert between square units, multiply by the appropriate conversion factor. For example, cm² to m²: multiply by 0.0001 (10⁻⁴).

The SA/V ratio is 2/r. This means thinner tubes (smaller r) have higher surface area relative to volume, which is important for heat exchange applications.

No, this calculator is specifically for tori (donut shapes). For spheroids, use an ellipsoid or sphere surface area calculator.

The cross-sectional area of the tube is πr², where r is the minor radius. This is the area of the circular cross-section at any point around the torus.

Surface area scales linearly with both R and r. Doubling R doubles the surface area; doubling r also doubles it.

The surface area equals the product of the two circumferences: the major circumference (2πR) and the minor circumference (2πr).

For a thin toroidal shell with inner and outer radii, you need to compute the difference between outer and inner torus surface areas.

The generating circle is the circle of radius r that is revolved around the axis at distance R from its center. Its revolution sweeps out the torus surface.

Surface area SA = 4π²Rr and volume V = 2π²Rr². The ratio SA/V = 2/r, independent of R.

A spindle torus has R < r, creating a self-intersecting shape. The standard formula SA = 4π²Rr still applies to the outer surface.

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