Torus Volume Calculator – Compute Doughnut Shape Volumes
A torus is a smooth, doughnut-shaped solid formed by rotating a circle around an external axis that lies in the same plane. It is defined by two measurements: the major radius R (distance from the center of the torus to the center of the tube) and the minor radius r (radius of the circular tube cross-section). This calculator computes the exact volume of any torus from those two radii, supports five input length units, and displays results in six volume units simultaneously.
The Volume Formula
The standard volume formula for a torus is:
V = 2π² × R × r²
This elegant result follows directly from Pappus's centroid theorem: the volume of any solid of revolution equals the area of the cross-section multiplied by the distance traveled by its centroid. For a torus, the cross-section is a circle of area πr², and its centroid travels a circumference of 2πR, giving V = πr² × 2πR = 2π²Rr². Because π² ≈ 9.8696, a quick mental approximation is V ≈ 19.739 × R × r².
Three Types of Torus
The relative sizes of R and r determine which type of torus you have:
- Ring Torus (R > r) — the familiar doughnut shape with a clearly visible hole running through the centre. This is the most common type encountered in everyday objects such as O-rings, bagels, and inner tubes.
- Horn Torus (R = r) — the inner hole collapses to a single point. The surface pinches at the inner equator and touches itself there, producing a shape that resembles a horn or pinched ring.
- Spindle Torus (R < r) — the tube is so thick that it overlaps itself, creating a self-intersecting shape sometimes called an apple-torus. The volume formula
V = 2π²Rr²still gives the correct algebraic volume, though the surface intersects itself.
The calculator automatically identifies which type you have entered and displays an appropriate label and description.
Supported Units
Input lengths can be entered in millimetres (mm), centimetres (cm), metres (m), inches (in), or feet (ft). The tool converts both radii to metres internally, computes the volume in m³, and then converts to your chosen output unit — cubic millimetres (mm³), cubic centimetres (cm³), cubic metres (m³), cubic inches (in³), cubic feet (ft³), or litres. Because volume scales as the cube of the linear dimension, the numeric values can differ dramatically between units — for instance, 1 m³ equals 1,000,000 cm³ or 1,000 litres.
Step-by-Step Formula Display
Below the result, the calculator shows the formula with all values substituted — for example:
V = 2 × π² × 10 × 3² = 2 × 9.8696 × 10 × 9 = 1776.53 cm³
This makes it straightforward to verify the calculation independently or include it in a technical report or assignment.
Real-World Applications
Torus volume calculations appear across a wide range of fields:
- Engineering — calculating the volume of O-ring seals, coil springs, toroidal pressure vessels, and ring-shaped fuel tanks used in spacecraft.
- Physics — modelling tokamak fusion reactor chambers, where plasma is confined in a toroidal magnetic field. The volume of the plasma chamber directly affects confinement calculations.
- Architecture and design — estimating material volumes for circular arch cross-sections, domed rooftop rings, and decorative toroidal features.
- 3D modelling and printing — generating physically accurate doughnut shapes, gaskets, and ring structures for CAD and additive manufacturing workflows.
- Mathematics education — illustrating Pappus's centroid theorem and surfaces of revolution in calculus courses.
Tips for Accurate Results
Both R and r must be entered in the same unit. If your measurements come from different sources in different units, convert them to a common unit before entering them here. Since the formula contains r², errors in the minor radius r are amplified — doubling r quadruples that factor in the result, so precise measurement of the tube radius is particularly important. For very large or very small values the calculator automatically switches to scientific notation to keep the display readable.