Torus Volume Calculator
Calculate the volume of a torus (donut shape) using V = 2*PI^2*R*r^2, where R is the distance from the center of the tube to the center of the torus and r is the tube radius. Used in engineering, physics, plasma confinement, and 3D modeling.
This free online torus volume calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Minimum: 0.01
Distance from the center of the torus to the center of the tube
Minimum: 0.01
Radius of the tube cross-section
Results
Volume
1421.223 cubic units
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Torus Volume Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Torus Volume Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Torus Volume Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Torus Volume Calculator
The Torus Volume Calculator computes the volume of a torus (donut shape) from its major radius R and minor radius r using V = 2*PI^2*R*r^2. A torus is the surface of revolution generated by rotating a circle of radius r about an axis at distance R from the circle's center. Tori appear in many contexts: donuts and bagels, inner tubes and tires, O-rings and gaskets, tokamak fusion reactors, and 3D computer graphics. The volume formula can be derived using Pappus' theorem or integration. This calculator serves engineers designing toroidal structures, physicists modeling plasma confinement, and students learning about solids of revolution.
The Math Behind It
Formula Reference
Torus Volume
V = 2 * pi^2 * R * r^2
Variables: R = major radius (center of torus to center of tube), r = minor radius (tube radius)
Worked Examples
Example 1: Standard Torus
A torus has major radius R = 8 and minor radius r = 3. Find its volume.
The volume is approximately 1421.22 cubic units.
Example 2: O-Ring
An O-ring has major radius 25 mm and tube radius 2 mm.
The O-ring volume is approximately 1973.92 cubic mm.
Common Mistakes & Tips
- !Confusing the major radius R (center of torus to center of tube) with the outer radius (R+r). The formula uses the center-to-center distance R.
- !Swapping R and r. R is the larger distance to the torus center; r is the smaller tube radius. The formula has R multiplied by r^2, so swapping gives a wrong answer.
- !Forgetting that PI appears squared in the formula. V = 2*PI^2*R*r^2, not 2*PI*R*r^2.
Related Concepts
Used in These Calculators
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Frequently Asked Questions
What is Pappus' theorem and how does it help?
Pappus' centroid theorem states that the volume of a solid of revolution equals the cross-sectional area times the distance traveled by the centroid of that cross-section. For a torus, the circular cross-section has area pi*r^2 and its centroid travels a circle of circumference 2*pi*R, giving V = 2*pi^2*R*r^2.
What is a horn torus?
A horn torus occurs when R = r (the tube just touches the center). A spindle torus occurs when R < r (the tube intersects itself through the center). The standard formula V = 2*pi^2*R*r^2 applies to all three types: ring torus (R > r), horn torus (R = r), and spindle torus (R < r).
Why are tokamak reactors shaped like a torus?
Magnetic field lines must form closed loops to confine plasma. A torus allows continuous field lines that wrap around the donut shape, keeping the hot plasma contained without hitting walls. This geometry is essential for achieving the conditions needed for nuclear fusion.
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