Torus Volume Calculator - Enclosed Volume in Any Unit

A torus volume calculator is a fast online tool that turns the major radius R and the minor radius r of a ring torus into its enclosed volume in real time, with the result expressed in the capacity unit you pick.

• • 3D printing and resin casting: Designers feeding a torus into a slicer or a mold-making pipeline get the exact filament or resin volume without re-deriving 2π²Rr².
• • Toroidal tank and pipe sizing: Engineers sizing a toroidal pressure vessel or swimming-pool tube read off the volume in liters and US gallons for capacity checks.
• • Cooking and baking conversions: Bakers scaling a donut-shaped cake, brioche, or bagel recipe convert the inner and outer dimensions of the mold into batter volume in cups and liters.
• • Geometry homework and exams: Students working through Pappus-centroid problems get an exact answer from 2π²Rr² with side-by-side unit readouts for sanity checks.

A torus is the doughnut shape you get by rotating a circle around an axis that sits in the same plane but does not touch the circle. Two numbers describe it completely: the major radius R and the minor radius r, and the enclosed volume follows from a 2D-to-3D theorem discovered by Pappus of Alexandria.

Because the volume scales with r², a small error in r becomes a big error in V. Pick the length unit (cm, m, in, or ft) for the linear inputs and the volume unit (cm³, m³, in³, ft³, L, US gal) for the capacity readout.

A sphere is a different solid, and its volume is given by V = (4/3)πr³, so for any spherical project the Sphere Volume Calculator is the right tool.

The calculator applies the standard ring-torus volume formula V = 2π²Rr², derives the surface area A = 4π²Rr, and reports inner and outer diameters and the aspect ratio so you can sanity-check the result.

• Major radius R: Distance from the center of the hole to the center of the tube. Sets the ring's overall diameter.
• Minor radius r: Radius of the tube that sweeps around the central axis. Sets the ring's thickness and is squared in the volume formula.
• Volume unit: The capacity unit used for the primary volume readout. The internal calculation runs in cubic centimeters, and the chosen volume unit is applied at the end with an exact factor.

Pappus's centroid theorem says that the volume of a solid of revolution equals the area of the generating shape times the distance traveled by its centroid. For a torus, the generating shape is a disk of area πr², and its centroid traces a circle of circumference 2πR. Multiplying them gives V = (πr²)(2πR) = 2π²Rr².

The same theorem applied to the generating circle (length 2πr) produces the surface area A = (2πr)(2πR) = 4π²Rr. Inner diameter = 2(R − r) and outer diameter = 2(R + r), so if you only know the inner and outer measurements you can solve for R and r and confirm the calculator is reading your shape correctly.

According to Wolfram MathWorld, the volume of a ring torus is V = 2π²Rr² and the surface area is A = 4π²Rr, where R is the major radius and r is the tube radius.

Like a cone, a torus is a solid of revolution produced by rotating a 2D shape around an axis, and the Cone Volume Calculator applies the same Pappus-style reasoning to a different generating area.

These four ideas cover every torus volume formula you will meet in a textbook, a CAD package, or a 3D printer slicer:

A useful intuition is to think of the torus as a cylinder of length 2πR and radius r bent into a circle. The squared r in the volume formula reflects the disk area that sweeps around the loop.

The clearest mental model for a torus is a bent cylinder of length 2πR and radius r. The Cylinder Volume Calculator is the right place to refresh that picture before applying the 2π²Rr² formula.

Four short steps take you from two raw measurements to a complete ring-torus capacity readout:

1. 1 Measure the major radius R: Find the distance from the center of the hole to the center of the tube. For a real object, average (outer Ø)/2 and (inner Ø)/2.
2. 2 Measure the minor radius r: Subtract the inner radius from the outer radius and divide by 2, or take the tube wall thickness from a spec. Use the same length unit as R.
3. 3 Pick a length unit and a capacity unit: Choose cm, m, in, or ft for the linear readouts, then choose cm³, m³, in³, ft³, L, or US gal for the primary volume readout. The other volume units stay visible for cross-checks.
4. 4 Read the result panel: Confirm the aspect ratio is greater than 1, then use the volume, surface area, and diameter readouts.

For a 3D-printed bracelet with R = 4 cm and r = 1 cm, the calculator reports about 78.96 cm³ of resin (0.079 L, 0.021 US gal), the right amount for a small ring-shaped mold.

If you need the same number in a unit that is not on the dropdown, the Volume Converter extends the result to teaspoons, tablespoons, cups, quarts, and many more capacity units.

Why reach for an online torus volume calculator instead of working the formula by hand?

• • Saves time on routine geometry: Replaces the manual steps of multiplying 2π², multiplying by R, squaring r, and reading the result with one input pair.
• • Capacity units side by side: Shows the same enclosed volume in cm³, L, and US gal at once, so the same number works for engineering specs, recipes, and fluid estimates.
• • Pairs volume with surface area: Reports A = 4π²Rr alongside V = 2π²Rr² so you can size a mold, coating, or thermal mass without leaving the page.
• • Catches shape mistakes: The aspect ratio and inner/outer diameter readouts immediately flag horn-torus and spindle-torus inputs that would break the ring-torus formula.
• • Length and capacity units stay independent: The length unit dropdown (cm, m, in, ft) governs R, r, and the inner/outer diameter readouts, while the volume unit dropdown (cm³, m³, in³, ft³, L, US gal) governs the capacity output, so the same shape can be sized in metric or imperial.

Because the calculator keeps R, r, the length-unit selector, and the volume-unit selector in their own fields, you can sweep through a parametric study by hand. Type a new R, watch the volume update, switch units and watch the readouts re-label.

For projects where the outer skin matters more than the interior fill, the Torus Surface Area Calculator emphasizes A = 4π²Rr and gives a different default aspect ratio.

The torus volume formula looks short, but the four factors below decide whether the answer matches the object in front of you:

• • The ring-torus volume formula assumes a continuous, smooth solid. A faceted polygon torus, a deformed tube, or a torus with a slot will deviate from the closed-form answer.
• • The volume returned is geometric. Real materials have porosity, coatings, and seams that change the true interior volume, so do not size paint or plating without a safety margin.

Another subtle factor is the difference between a torus and a solid torus. The formulas on this page cover the solid doughnut you get by rotating a full disk, so the enclosed volume V is the solid-torus volume.

According to Wikipedia, the volume of a ring torus is V = (πr²)(2πR) = 2π²Rr² and the surface area is A = (2πr)(2πR) = 4π²Rr, where R is the distance from the center of the tube to the center of the torus and r is the radius of the tube. Encyclopaedia Britannica describes a torus as a surface of revolution generated by rotating a circle about an axis in its plane that does not intersect the circle.

If you need to compare the torus volume with a sphere, cone, or cylinder in the same design, the Volume Calculator gives you side-by-side volume calculations without leaving Math & Conversion.