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

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Last updated: 2026-05-23 · Reviewed by Nham Vu