Torus Volume and Surface Area Calculator

A torus is a donut shape, a circular tube bent around into a ring. It is defined by two numbers: a major radius R from the center of the hole to the center of the tube, and a minor radius r for the tube cross section itself.

Volume equals 2 times pi squared times the major radius times the minor radius squared, or 2 pi squared R r squared. It follows from Pappus's theorem: the tube's circular cross section area pi r squared, swept around the ring of circumference 2 pi R.

Surface area equals 4 times pi squared times the major radius times the minor radius, or 4 pi squared R r. That is the tube circumference 2 pi r multiplied by the path length 2 pi R its center traces around the ring.

The major radius R is the distance from the center of the hole to the center of the tube. The minor radius r is the radius of the tube's circular cross section. The outer edge of the donut sits at R plus r, and the hole edge at R minus r.

Volume is 2 pi squared R r squared and surface area is 4 pi squared R r, so they are equal whenever r squared times 2 equals 4 r, that is when r equals 2, for any R. At R 5 and r 2 both work out to 394.784176 in their respective units.

A standard ring torus keeps R at least as large as r so the tube does not overlap itself. This calculator accepts any positive values, but when r is larger than R the shape self-intersects and the formulas describe the swept solid rather than a clean donut.

Each radius must be greater than zero and can range up to 1000, adjusted in steps of 0.1. The units are whatever you use for the radii: volume comes out in those units cubed and surface area in those units squared, so radii in cm give volume in cubic cm.

The engine uses JavaScript's full double-precision value of pi, then rounds each answer to six decimal places. Displayed values group thousands with a comma, so 1776.528792 appears as 1,776.528792.

No. The whole calculation runs locally in your browser, so the radii you type and the results are never uploaded to a server. You can use the tool offline once the page has loaded.