Torus Volume Calculator
Enter the centre radius and tube radius of a torus — a ring or doughnut — for its volume and surface area, drawn to scale. Click a diameter on the drawing to change it.
Dimensions
TorusResult
Volume of a torus
A torus is a circle swept around an axis it does not touch — a ring. Two radii set it: R from the ring centre to the tube centre, and a, the radius of the tube. Volume and surface area both follow from Pappus's theorem.
Editing on the drawing
Click the centre diameter or the tube diameter on the figure and type a new value to redraw the ring to scale.
How to find the volume of a torus
A torus is the ring you get by sweeping a circle all the way around an axis it doesn't cross — the shape of a doughnut or an O-ring. Two radii describe it: R, from the centre of the ring to the centre of the tube, and a, the radius of the tube itself.
The formulas
Both formulas come from Pappus's theorem: the swept area (or its perimeter) times the distance the centroid travels, 2πR.
Worked example
Take a torus with centre radius 64 and tube radius 22.
Tube area times the path of its centre.
Tube circumference times that path.
What else the calculator returns
You also get the outer diameter 2(R + a), the inner (hole) diameter 2(R − a) and the tube diameter 2a. The ring stays a clean doughnut only while the tube radius is smaller than the centre radius.
Where it comes up
Torus volumes describe O-rings and seals, tyres and inner tubes, doughnuts, wound coils and architectural rings.
Frequently asked questions
How do you find the volume of a torus?
Use the two radii: V = 2π²Ra², where R is the centre radius and a the tube radius.
What do R and a mean?
R is the distance from the centre of the ring to the centre of the tube; a is the radius of the tube's circular cross-section.
What is Pappus's theorem?
It states that a solid of revolution's volume is the swept area times the distance its centroid travels — here πa² × 2πR — which gives the torus formulas directly.
What happens if the tube radius equals the centre radius?
When a equals R the hole shrinks to a point; beyond that the tube overlaps itself and the simple formula no longer describes a clean ring.
How big is the hole in the middle?
The inner (hole) diameter is 2(R − a), while the outer diameter is 2(R + a).