Torus Volume Calculator
A torus is the three-dimensional shape you get by rotating a circle around an axis that lies in the same plane as the circle but does not pass through it. The result looks like a donut, a ring, or an inner tube. Two measurements fully describe a torus: the major radius R, measured from the centre of the whole shape to the centre of the tube, and the minor radius r, which is the radius of the circular cross-section of the tube itself. A ring with a large hole and thin tube has R much larger than r. A plump donut shape has R and r closer in magnitude. If R equals r the inner hole vanishes entirely and the shape is called a horn torus; if R is less than r the tube self-intersects. Toroidal shapes appear in engineering for O-rings, gaskets, and magnetic field coils, in physics for toroidal plasma chambers in fusion reactors, and in mathematics as one of the simplest curved surfaces. This calculator takes R and r and returns the volume, the total surface area, the outer diameter, the inner diameter (the hole diameter), and the tube cross-section area. The default uses R = 5 and r = 2, giving volume = 394.78 and surface area = 394.78 (a coincidence for these specific values).
Volume = 2π²Rr². Surface area = 4π²Rr. R must be greater than r for a standard ring torus.
How it works
For major radius R and tube radius r: volume = 2 * pi^2 * R * r^2. This comes from Pappus's centroid theorem: the volume of a solid of revolution equals the cross-sectional area (pi * r^2) times the distance the centroid travels (2 * pi * R). Surface area = 4 * pi^2 * R * r, likewise the perimeter of the cross-section (2 * pi * r) times the centroid path. Outer diameter = 2 * (R + r). Inner diameter = 2 * (R - r). Tube cross-section area = pi * r^2.
Worked example
For R = 5 and r = 2: volume = 2 * pi^2 * 5 * 4 = 40 * pi^2 = 394.78. Surface area = 4 * pi^2 * 5 * 2 = 40 * pi^2 = 394.78. Outer diameter = 2 * 7 = 14.00. Inner diameter = 2 * 3 = 6.00. Tube cross-section = pi * 4 = 12.57. These match the defaults pre-filled above. The equality of volume and surface area for R = 5 and r = 2 is a numerical coincidence.
Related calculators
- Hemisphere Calculator: volume and surface area of a hemisphere.
- Frustum Calculator: volume and surface area of a truncated cone.
- Annulus Calculator: the flat ring area that forms the torus cross-section.
- Circle Segment Calculator: area and arc of a circular segment.