Torus Surface Area Calculator

Calculate the surface area of a torus from its major radius R and minor radius r. Also shows inner and outer radii.

Enter Torus Dimensions

Result

Surface Area
--
square units
Outer Radius (R + r)--
Inner Radius (R - r)--
Overall Diameter--
Tube Circumference--

Step-by-Step Solution

SA = 4π²Rr

Understanding the Torus

A torus is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle that does not intersect it. It is defined by two radii: the major radius R (distance from the center of the tube to the center of the torus) and the minor radius r (radius of the tube itself).

Torus Formulas

Surface Area

The total outer surface area of the torus.

SA = 4π²Rr

Volume

The amount of space enclosed by the torus.

V = 2π²Rr²

Outer Radius

The maximum distance from the center of the torus to its outer edge.

R_outer = R + r

Inner Radius

The minimum distance from the center of the torus to its inner edge (hole).

R_inner = R - r

Conditions for a Valid Torus

  • Ring torus: R > r. This is the standard doughnut shape with a hole in the center.
  • Horn torus: R = r. The inner radius is zero and the torus just touches itself at the center.
  • Spindle torus: R < r. The torus self-intersects, creating a different topology.

Applications

Tori appear in many areas of mathematics and engineering, including topology, differential geometry, plasma physics (tokamak fusion reactors), and industrial design (O-rings, inner tubes, and bagels). The surface area formula is essential for calculating material costs and heat transfer rates on toroidal surfaces.