Torus Volume Calculator (Donut Volume)

A torus is a donut shape, a circle swept around an axis to form a ring. It is defined by two radii:

• R (major radius): distance from the center of the ring to the center of the circular cross-section.
• r (minor radius): the radius of the circular cross-section (“the tube radius”).

V = 2 × π² × R × r²

You can also derive this via Pappus’s theorem: the volume of a solid of revolution equals the cross-sectional area times the distance traveled by the centroid. Cross-section area is πr², centroid travels a circle of circumference 2πR, so V = πr² × 2πR = 2π²Rr².

Worked example: bicycle inner tube A 700×25c road tire takes a tube sitting at roughly R ≈ 33 cm (the wheel’s mean radius) with r ≈ 1 cm once it has filled out a 25 mm casing. V = 2 × π² × 33 × 1² = 66π² ≈ 651 cm³.

That is the space inside the tube. It is not the amount of air you have to push in, and mixing those two up is the classic error here. Air compresses. At 100 psi on the gauge the tube is holding air at about 7.8 times atmospheric pressure ((100 + 14.7) ÷ 14.7), so inflating it from flat takes roughly 651 × 7.8 ≈ 5,100 cm³ of free air.

A decent floor pump moves something like 400 to 800 cm³ per stroke, which puts that tire at maybe 7 to 13 strokes. A mini frame pump shifting 60 cm³ a stroke needs about 85 of them, which is precisely why nobody enjoys using one at the roadside.

Where torus volumes matter:

• Inner tubes for bikes, cars, trucks.
• Rubber O-rings. Tiny donut-shaped seals between mechanical parts. Engineers calculate O-ring squeeze (deformation under compression) using the torus geometry.
• Donuts and bagels. A glazed donut is roughly R = 3 cm, r = 1.5 cm: V ≈ 133 cm³. A bagel has the same shape but bigger and denser.
• Magnetic-confinement fusion reactors (tokamak design). The plasma is shaped as a torus.
• Tire tubes for inflatable rafts and ring buoys.
• Architectural rings. Toroidal arches and curved structural members.

Pappus’s theorem, and the intuition behind it:

Imagine slicing the donut along its major axis to get a circle of area πr². Rotate that circle through 360° around the central axis and every point in it traces a circle in 3D space. The average point, the centroid, traces a circle of radius R with circumference 2πR. So the volume is (cross-section area) × (distance traveled by the centroid) = πr² × 2πR = 2π²Rr².

Another way to see the same thing: straighten the tube out. You get a cylinder 2πR long with a circular cross-section of area πr², and its volume is the same number. That is why the related hollow-cylinder page and this one agree once you unbend the ring.

This trick generalizes. ANY shape revolved around an external axis has volume equal to that shape’s area times the path length its centroid travels. Pappus was right, and it is a beautiful piece of geometry.

Limit cases:

• R = 0: the ring radius vanishes and the formula gives 0.
• r = 0: V = 0 (a zero-thickness ring).
• R = r: the hole closes down to a single point. This is a “horn torus”, and the formula is still exact, because the generating circle only touches the axis instead of crossing it.
• R < r: the generating circle crosses the axis and the swept solid overlaps itself, which is a “spindle torus”. Here 2π²Rr² double-counts the overlap, so it stops being the volume of the solid. The calculator flags this case rather than printing the number without comment.

Surface area is a separate quantity: SA = 4π² × R × r (covered on the torus surface area page).

Sanity check:

• For R = 5, r = 1: V = 2π² × 5 × 1 = 10π² ≈ 98.7 cubic units.
• Doubling r → 4× the volume (quadratic).
• Doubling R → 2× the volume (linear).