Torus Volume Calculator
Find the volume and surface area of a torus.
Center to tube center
Tube radius
Results update as you type.
How it works
A torus is the mathematical name for a donut or ring shape, and this calculator finds its volume and surface area. You enter the major radius, the distance from the center of the hole to the center of the tube, and the minor radius, the radius of the tube itself, and it returns the volume and the surface area. The formulas come from Pappus's centroid theorem, which relates the volume and surface of a solid of revolution to the shape being revolved and the distance its centroid travels. Revolving a circle of radius r around an axis at distance R produces a torus, giving a volume of two pi squared times R times r squared and a surface area of four pi squared times R times r. These compact formulas make the torus one of the more elegant solids to measure. Tori appear widely: in doughnuts and bagels, O-rings and gaskets, inner tubes and life rings, magnetic confinement devices like tokamaks, and architectural and design elements. Computing their volume and surface helps with material estimates, capacity, and engineering. For a proper torus with a hole, the minor radius must be smaller than the major radius; if the minor radius equals or exceeds the major, the shape self-intersects and the simple formulas no longer describe a standard ring. Enter the major and minor radii to compute the torus's volume and surface area.
V = 2π²Rr² ; Surface area = 4π²Rr
A torus is a donut shape defined by a major radius R, from the center to the middle of the tube, and a minor radius r, the tube's own radius. By Pappus's theorem, the volume is 2π²Rr² and the surface area is 4π²Rr, obtained by revolving the tube's circle around the central axis.
Source:Solid geometry (Pappus's theorem)
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter major radius (r) and minor radius (r).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Volume updates instantly as you type — no button to press.
Frequently asked questions
What is a torus?
A torus is a donut-shaped surface or solid formed by revolving a circle around an axis that lies in the same plane but does not touch the circle. It is described by two radii: the major radius from the center of the hole to the center of the tube, and the minor radius of the tube itself. Rings, inner tubes, and bagels are everyday tori.
How do you find the volume of a torus?
The volume of a torus is two pi squared times the major radius times the minor radius squared. This comes from Pappus's theorem, which gives the volume of a solid of revolution as the area of the revolved shape times the distance its centroid travels. The calculator applies this formula from your two radii.
What are the major and minor radii?
The major radius, R, is the distance from the center of the torus's hole to the center of the circular tube. The minor radius, r, is the radius of the tube's cross-section. Together they define the size and thickness of the ring: a large R with a small r gives a thin, wide ring, while a small R with a larger r gives a fat, compact one.
Why must the minor radius be smaller than the major?
For a standard torus with an open hole in the middle, the tube must not reach the central axis, which requires the minor radius to be less than the major radius. If they are equal the hole closes to a point, and if the minor radius is larger the surface self-intersects, so the simple volume and surface formulas no longer describe an ordinary ring shape.
Where are torus calculations used?
Tori model O-rings and gaskets in engineering, inner tubes and life rings, doughnuts and bagels in food, coils and transformers in electronics, and fusion reactors called tokamaks. Calculating volume and surface area helps estimate material, capacity, and coatings for these ring-shaped objects, making the torus a practically important as well as elegant shape.