Quick answer
A torus with major radius R and minor radius r has volume 2π²Rr² and surface area 4π²Rr. With R = 3 and r = 1 the volume is 59.2176 and the surface area 118.4353.
Enter the major and minor radii. The calculator returns volume and surface area.
Related tools: Cylinder Volume Calculator: Volume of a cylinder. Sphere Volume Calculator: Volume and surface area of a sphere. Volume of a Hemisphere Calculator: Volume and surface area of a hemisphere.
How to use the Torus Volume Calculator
- Enter major radius (above 0).
- Enter minor radius (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Major radius | A number | > 0 |
| Minor radius | A number | > 0 |
Formula and method
Formula
V = 2π²Rr²; A = 4π²Rr
IEEE-754 double precision; shown to at most 6 decimal places.
Pappus's centroid theorem; requires r ≤ R for a ring torus.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Major radius = 3; Minor radius = 1.
| Result | Value |
|---|---|
| Volume | 59.217626 |
| Surface area | 118.435253 |
Example 2
Inputs: Major radius = 2; Minor radius = 2.
| Result | Value |
|---|---|
| Volume | 157.91367 |
| Surface area | 157.91367 |
The first line is the headline result (volume). Below it the torus Volume Calculator lists surface area, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Pappus's theorem
The volume equals the area of the circle (πr²) times the distance its centre travels (2πR): π × 1 × 6π = 59.22. The surface area is the circle's circumference (2πr) times the same distance.
Condition
The minor radius must not exceed the major radius, or the ring self-intersects. Doughnuts, tyre inner tubes and O-rings are tori.
How the result changes with major radius
Holding every other input at the first worked example, this table changes only major radius. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Major radius | Volume | Surface area |
|---|---|---|
| 1.5 | 29.608813 | 59.217626 |
| 3 | 59.217626 | 118.435253 |
| 6 | 118.435253 | 236.870506 |
| 12 | 236.870506 | 473.741011 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Minor radius: entering "2" shows "The minor radius must not exceed the major radius." beside that field and no result.
Choosing between the Torus Volume Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Cylinder Volume Calculator](/math/cylinder-volume-calculator/) | radius and height | Volume |
| [Sphere Volume Calculator](/math/sphere-volume-calculator/) | radius | Volume |
| [Volume of a Hemisphere Calculator](/math/volume-of-hemisphere/) | radius | Volume |
| [Cone Volume Calculator](/math/cone-volume-calculator/) | radius and height | Volume |
| [Pyramid Volume Calculator](/math/pyramid-volume-calculator/) | base length, base width and height | Volume |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Minor radius ≤ major radius
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for minor radius to confirm this.
Frequently asked questions
What is R?
The distance from the centre of the hole to the centre of the tube.
What is r?
The radius of the tube.
How do I find the volume of an O-ring?
Use the tube's radius as the minor radius and the distance from the ring's centre to the tube centre as the major radius.
What does the Torus Volume Calculator calculate?
Volume and surface area of a torus. It returns Volume and Surface area.
What formula does the Torus Volume Calculator use?
V = 2π²Rr²; A = 4π²Rr. Pappus's centroid theorem; requires r ≤ R for a ring torus.
What do I need to enter in the torus Volume Calculator?
You enter Major radius and Minor radius.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Minor radius ≤ major radius.
What is an example of a torus Volume result?
With major radius 3 and minor radius 1, the volume is 59.217626.
Related calculators
- Cylinder Volume Calculator — Volume of a cylinder.
- Sphere Volume Calculator — Volume and surface area of a sphere.
- Volume of a Hemisphere Calculator — Volume and surface area of a hemisphere.
- Cone Volume Calculator — Volume, slant height and surface area of a cone.
- Pyramid Volume Calculator — Volume of a pyramid from its base area and height (rectangular base).
- Cube Volume Calculator — Volume, surface area and diagonal of a cube.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.