Torus Calculator

A torus is a 3D geometric shape formed by revolving a circle around an axis that lies in the same plane as the circle but does not intersect it. Common real-world examples include doughnuts, ring-shaped life preservers, and inflatable tire tubes.

A torus is created by taking a circle with radius r (the minor radius) and rotating it around an external axis at a distance R (the major radius) from the circle's center. The result is a smooth, ring-shaped surface enclosing a volume.

There are three types based on the relationship between R and r: a Ring Torus (R > r) has a hole in the middle; a Horn Torus (R = r) has the inner surface meeting at a single point at the center; and a Spindle Torus (R < r) self-intersects. This calculator supports ring and horn tori.

The volume of a torus is calculated using the formula V = 2π² × R × r², where R is the major radius (center of torus to center of tube) and r is the minor radius (radius of the tube). Alternatively, using inner radius a and outer radius b: V = (π²/4) × (a + b) × (b − a)².

The surface area of a torus is given by A = 4π² × R × r, where R is the major radius and r is the minor radius. This formula represents the total outer surface of the torus.

The inner radius (a) is the distance from the center of the torus to the nearest edge of the tube, calculated as a = R − r. The outer radius (b) is the distance from the center to the farthest edge, calculated as b = R + r.

The calculator uses generic units — whatever unit you use for R and r (e.g. centimeters, meters, inches), the volume result will be in cubic units of that measurement and the surface area in square units. Just make sure both R and r use the same unit.

Yes. Any ring-shaped or toroidal object — including doughnuts, O-rings, tires, and circular tubes — can be modeled as a torus. Simply measure or estimate the major radius R (center-to-tube-center) and minor radius r (tube radius) and enter them into the calculator.