Torus Calculator

Torus Calculator. Enter the major radius (R) — the distance from the center of the torus to the center of the tube — and the minor radius (r) — the radius of the tube itself — to calculate the volume and surface area of a torus. You also get the inner radius, outer radius, and torus type (ring, horn, or spindle) based on your inputs. Also try the find Congruence Result with Congruent Triangles Calculator.

Results

Volume

Surface Area

Outer Radius (b = R + r)

Inner Radius (a = R − r)

Torus Type

What is a torus?

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. See also our calculate Surface Area of a Rectangular Prism.

How is a torus formed?

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.

What are the different types of tori?

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.

How do you calculate the volume of a torus?

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.

How do you calculate the surface area of a torus?

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.

What is the difference between inner radius and outer radius?

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.

What units does this torus calculator use?

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.

Can this calculator be used for doughnut or tire calculations?

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. You might also find our Circle Calculator useful.