Torus Calculator — Volume, Surface Area and Radii

Calculate a ring torus from its major and minor radii, inner and outer radii, or inner and outer diameters.

Torus dimensions

Known measurements R reaches the tube center; r is the radius of the tube.

Major radius (R) From the torus center to the tube center.

Minor radius (r) The circular tube radius.

Length unit

Decimal places

π value

Your measurements stay in this browser. Enter both lengths in one common unit; the selector labels results but does not convert mixed inputs.

Results

Enter two torus measurements.

Your volume, surface area, radii, diameters, and formula substitutions will appear here.

Torus Formulas

This calculator models a circular tube revolved around an axis in the same plane. The major radius R is the distance from the rotation axis to the center of the tube. The minor radius r is the radius of the tube itself.

Measurement	Formula	Meaning
Volume	`V = 2π²Rr²`	Space enclosed by the torus
Surface area	`A = 4π²Rr`	Area of the entire curved surface
Outer radius	`a = R + r`	Center to the farthest outside edge
Inner radius	`b = R − r`	Center to the nearest edge of the hole
Tube cross-section area	`A_c = πr²`	Area of a cut through the circular tube
Centerline circumference	`C = 2πR`	Distance traveled by the tube center in one revolution

Units: radii and diameters use length units, surface and cross-section areas use squared units, and volume uses cubed units. Values are rounded only for display after the formulas are evaluated at full JavaScript floating-point precision.

Worked Example

Volume and surface area

For R = 5 cm and r = 2 cm, volume is 40π² ≈ 394.78 cm³ and surface area is 40π² ≈ 394.78 cm².

Inner and outer sizes

The same torus has outer radius 7 cm, inner radius 3 cm, outer diameter 14 cm, and inner diameter 6 cm.

How to Measure a Torus

If you can locate the center of the tube, measure from the torus center to that tube center for R, then measure the tube radius r.
If the centerline is hard to locate, measure the farthest and nearest distances from the torus center and choose Outer radius and inner radius.
If measuring all the way across is easier, use the full outside and hole widths and choose Outer diameter and inner diameter.
Keep both measurements in the same unit before calculating.

A physical torus may have an elliptical tube, flattened surfaces, seams, or manufacturing tolerances. This page assumes an ideal torus with a perfectly circular tube and rotational symmetry.

Ring Torus, Horn Torus, and Input Limits

Ring torus: R > r. The center hole is open and its radius is R − r.
Horn torus: R = r. The hole closes to one point, so the inner radius is zero.
Spindle torus: R < r. Its generating circle crosses the rotation axis and the surface self-intersects; this calculator rejects that case.
Positive measurements from 10⁻¹⁰⁰ through 10¹⁰⁰ are accepted. An inner radius or diameter may be zero for a horn torus.

Methodology and Assumptions

The volume can be understood with Pappus’s centroid theorem: the tube’s circular cross-section, area πr², travels a centerline distance 2πR, giving V = 2π²Rr². The analogous surface result is the tube circumference 2πr times that centerline distance, giving A = 4π²Rr.

References: formulas cross-checked with Wolfram MathWorld: Torus and the OpenStax derivation of torus volume using Pappus’s theorem. Last reviewed: July 30, 2026.

Torus Calculator FAQs

What is the volume formula for a torus?

For major radius R and minor radius r, the volume of a ring torus is V = 2π²Rr².

What is the surface area formula for a torus?

For major radius R and minor radius r, the surface area is A = 4π²Rr.

What are the major and minor radii of a torus?

The major radius R runs from the torus center to the center of its circular tube. The minor radius r is the radius of that tube.

How do I find torus radii from the outer and inner radii?

If a is the outer radius and b is the inner radius, then R = (a + b)/2 and r = (a − b)/2.

Can I calculate a torus from its outer and inner diameters?

Yes. For outer diameter D_o and inner diameter D_i, R = (D_o + D_i)/4 and r = (D_o − D_i)/4.

What dimensions make a ring torus?

A ring torus has R > r, leaving an open central hole. When R = r, the limiting shape is a horn torus with zero inner radius.

Does this calculator convert between units?

No. Enter both measurements in one common length unit. The calculator labels lengths, areas, and volume with the corresponding unit powers.

Are my torus measurements uploaded?

No. The calculation runs locally in your browser and the page does not send your measurements to a server.