Torus Volume Calculator (Donut Volume)

A torus is a donut shape — a circle swept around an axis to form a ring. It’s 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 bike tire holds an inner tube with roughly R ≈ 33 cm (mean wheel radius) and r ≈ 1 cm (when inflated to 100 psi for a 25 mm tire). V = 2 × π² × 33 × 1 = 65 × π² ≈ 642 cm³ ≈ 0.64 L of air at atmospheric pressure.

At 100 psi (about 7 bar absolute), the same air mass compresses to roughly 642 / 7 ≈ 92 cm³ of inflated tube. Sounds about right — a bike pump cranks ~60-100 cm³ per stroke and most road tires need maybe 8-15 strokes from flat.

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 — the intuition:

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

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; it’s a beautiful piece of geometry.

Limit cases:

• R = 0: shape collapses, formula gives 0. (A torus where the hole has zero size is no longer a torus.)
• r = 0: V = 0 (zero-thickness ring).
• R = r: still works, but the torus self-intersects at the inner edge (“horn torus”). The formula gives the “envelope” volume — what you’d get filling a mold of the swept shape.
• R < r: the torus self-intersects into a “spindle torus” — the formula is no longer geometrically clean.

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).