What is the formula for the surface area of a torus?

The total surface area of a torus is SA = 4π²Rr, where R is the major radius (center of torus to center of tube) and r is the minor radius (radius of the tube). This result comes from Pappus's centroid theorem applied to a circle of circumference 2πr revolved around an axis at distance R.

What is the difference between major radius R and minor radius r?

R (major radius) is the distance from the center of the torus to the center of the tube — essentially how large the ring is overall. r (minor radius) is the radius of the tube's circular cross-section — how thick the ring is. For a valid (non-self-intersecting) torus, R must be greater than or equal to r.

What happens when R equals r?

When R = r the torus degenerates into a horn torus, where the inner hole shrinks to a single point. The surface area formula SA = 4π²r² still applies in this limiting case.

What units does the calculator use?

The calculator works with any consistent unit. If you enter R and r in centimeters, the area is returned in cm². If you use meters, the result is in m². Just make sure both radii use the same unit.

Can this be used for a ring torus, horn torus, and spindle torus?

The formula SA = 4π²Rr correctly gives the outer surface area for a ring torus (R > r). For a horn torus (R = r) the formula still holds. A spindle torus (R < r) self-intersects, so its surface area calculation is more complex; this calculator flags that case as invalid.

Surface Area of a Torus Calculator

Name: Surface Area of a Torus Calculator
Availability: InStock
Author: Nham Vu

Enter the major radius R and tube radius r to instantly compute the total surface area of a torus using the formula 4π²Rr.

Torus Dimensions

Enter both radii in the same unit (mm, cm, m, in, ft, …).

Major Radius R (center of torus to center of tube)

Minor Radius r (radius of the tube)

Unit label (optional, for display only)

Formula

SA = 4π² × R × r

Derived from Pappus's centroid theorem

Result

Enter R and r, then click Calculate.

Torus Diagram

R — major radius: distance from the center of the torus to the center of the tube.

r — minor radius: radius of the circular cross-section of the tube.

The surface is generated by rotating the circle of radius r around the central axis at distance R.

SA = 4π² × R × r

Summary

Enter the major radius R and tube radius r to instantly compute the total surface area of a torus using the formula 4π²Rr.

How it works

Enter the major radius R — the distance from the center of the torus to the center of the tube.
Enter the minor radius r — the radius of the circular tube cross-section.
Both radii must be positive and R must be greater than or equal to r to form a valid torus.
The calculator applies the formula SA = 4π²Rr and displays the result.
Copy the result or adjust the inputs to explore different torus dimensions.

Use cases

Engineering design of toroidal pressure vessels, gaskets, and O-rings.
Architecture and art installations featuring torus-shaped structures.
Physics and mathematics coursework involving surface integrals.
Calculating material coverage for coating or painting a torus surface.
Manufacturing cost estimation for ring-shaped products.
3D printing and CAD modeling of toroidal geometry.
Teaching geometry concepts about solids of revolution.
Scientific research on toroidal shapes in plasma physics and electromagnetics.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu