A torus is a ring-shaped solid like a doughnut, an inner tube or an O-ring. Enter the major radius R and the tube radius r to get the volume and the surface area. The page also reports the hole radius and the outer radius.
24 solids in six groups. Enter the dimensions and the model updates instantly.
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R is the distance from the centre of the hole to the centre of the tube, and r is the radius of the tube itself. R must be larger than r, or the ring would overlap itself. Use Diameter mode if you measured across the tube.
This page is the torus view of the full 3D Shape Calculator, which covers 24 solids. Every shape button is still available, so you can compare solids without leaving the page.
In every formula, r is a radius, h is a height or depth, and all measurements must use the same unit. The result is in that unit cubed for volume and squared for area.
A bicycle inner tube has R = 30 cm and a tube radius of 5 cm. Its volume is 2 × π² × 30 × 5² ≈ 14,804 cm³, about 14.8 litres of air, and its outer surface is 4 × π² × 30 × 5 ≈ 5,922 cm² of rubber.
Tori describe doughnuts, inner tubes, O-rings, lifebuoys, ring magnets, toroidal transformer cores and the shape of some particle accelerators and fusion reactors.
What is the volume of a torus? V = 2 × π² × R × r², where R is the major radius and r is the tube radius.
Where does the formula come from? It comes from the theorem usually credited to Pappus: the volume of a solid of revolution equals the area of the shape that is turned, here π × r², multiplied by the distance its centre travels, 2 × π × R.
Why must R be larger than r? If R were smaller than r the tube would cross the axis and overlap itself, which no longer describes a ring. The calculator warns you when that happens.
How are volume and surface area related? Dividing the volume by the surface area gives r / 2 for any torus, a neat check on your numbers.