A torus is the three-dimensional solid formed by rotating a circle around an axis in its own plane that does not intersect the circle. It looks like a donut or an inflatable ring. It is defined by two radii: R is the major radius from the centre of the torus to the centre of the tube, and r is the minor radius (tube radius), the radius of the circular cross-section.

What is the volume formula for a torus?

The volume of a torus is 2 pi squared times R times r squared, where R is the major radius and r is the tube radius. This formula comes from Pappus's centroid theorem: the volume of a solid of revolution equals the area of the cross-section (pi r squared) times the distance the centroid travels (2 pi R).

What is the surface area formula for a torus?

The surface area of a torus is 4 pi squared times R times r. Again this follows from Pappus's theorem: the surface area equals the perimeter of the cross-section circle (2 pi r) times the distance the centroid travels (2 pi R), giving 4 pi squared R r. For the default values R = 5 and r = 2, both volume and surface area equal approximately 394.78.

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).

Major radius (R)

units

Tube radius (r)

units

394.78

volume (cubic units)

Surface area394.78

Outer diameter14.00

Inner diameter (hole)6.00

Tube cross-section12.57

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.

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