Torus Volume Calculator

Calculate the volume of a torus using its major and minor radii.

A torus is a 3D shape formed by revolving a circle around an axis in its plane. This tool helps you find its volume with high precision.

Examples

Click on an example to load its data into the calculator.

Standard Torus

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A standard torus with a large major radius and a smaller minor radius.

R: 10

r: 3

Thick Ring (O-Ring)

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An example representing a thick ring or an O-ring, where the minor radius is close to the major radius.

R: 5

r: 2

Large, Thin Tube

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A torus representing a large, thin circular tube, like an inner tube for a large tire.

R: 20

r: 2

Small Decorative Ring

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A small torus, similar in proportion to a decorative ring or a donut.

R: 4

r: 1.5

Other Titles
Understanding the Torus Volume Calculator: A Comprehensive Guide
Explore the mathematics behind torus volume, its real-world applications, and how to use this calculator effectively.

What is a Torus? Foundations and Key Concepts

  • A torus is a surface of revolution generated by revolving a circle in three-dimensional space.
  • It is defined by two key parameters: the major radius (R) and the minor radius (r).
  • The shape is commonly known as a donut or a ring.
A torus is a fundamental geometric shape that appears frequently in mathematics, physics, and engineering. It's a surface of revolution created by rotating a circle (the 'minor' circle) around an axis that lies in the same plane as the circle but does not intersect it. The result is a donut-like shape.
Defining Radii
To understand and calculate the properties of a torus, two radii are essential: Major Radius (R) is the distance from the center of the entire torus to the center of the tube, defining its overall size. Minor Radius (r) is the radius of the revolving circle itself, defining the thickness of the ring.
A critical constraint is that the major radius (R) must be greater than the minor radius (r). If R=r, it forms a 'horn torus' with no central hole. If R<r, the torus self-intersects, creating a 'spindle torus'.

Step-by-Step Guide to Using the Torus Volume Calculator

  • Enter the major and minor radii into the designated fields.
  • Click the 'Calculate' button to compute the volume.
  • Review the result and use the 'Reset' button for new calculations.
Our calculator simplifies the process of finding the volume of a torus. Follow these simple steps for an accurate result.
Input Instructions:
1. Enter Major Radius (R): In the first input field, type the value for the major radius. This must be a positive number. 2. Enter Minor Radius (r): In the second input field, type the value for the minor radius. This must also be a positive number and must be less than the major radius. 3. Calculate: Press the 'Calculate Volume' button.
Interpreting the Output:
The result displayed is the volume (V) of the torus in cubic units, corresponding to the units of the input radii. You can easily copy the result to your clipboard using the copy icon.

Practical Usage Examples

  • Input: R=10, r=2 → Volume ≈ 789.57
  • Input: R=6, r=5.5 → Volume ≈ 3581.08

The Mathematical Formula and Derivation

  • The volume of a torus is calculated using the formula V = 2π²Rr².
  • This formula can be derived using Pappus's second centroid theorem.
  • It represents the area of the minor circle multiplied by the circumference of the path traced by its centroid.
The elegance of the torus volume formula lies in its simplicity and the geometric principles it represents. The standard formula is: V = (πr²) * (2πR) = 2π²Rr².
Derivation using Pappus's Theorem
Pappus's second centroid theorem states that the volume of a solid of revolution is the product of the area of the generating figure (A) and the distance (d) traveled by its geometric centroid. For a torus, the generating figure is a circle with radius 'r' (Area = πr²), and its centroid travels a distance of d = 2πR.
Multiplying these two together gives the volume: V = A d = (πr²) (2πR) = 2π²Rr².

Real-World Applications of the Torus

  • Engineering applications like O-rings, seals, and gaskets.
  • Physics, particularly in magnetic confinement for fusion reactors (tokamaks).
  • Architecture and design for creating unique, curved structures.
The torus shape is not just a mathematical curiosity; it is integral to many practical applications.
Engineering and Manufacturing
In mechanical engineering, toroidal shapes are found in O-rings, which are used as seals, and in the design of pipes and pressure vessels. Their continuous, smooth surface makes them ideal for creating tight seals.
Physics and Energy
Perhaps one of the most famous applications is the tokamak, a device that uses a magnetic field to confine plasma in the shape of a torus. This is a leading approach for achieving controlled thermonuclear fusion power.
Computer Graphics and Design
In 3D modeling and computer graphics, the torus is a primitive shape used to create more complex objects, from lifebuoys to architectural elements.

Industry Applications

  • Designing an O-ring with a specific volume for a hydraulic system.
  • Calculating the plasma volume in a conceptual tokamak design.
  • Estimating material needed for a toroidal architectural feature.

Common Questions and Advanced Topics

  • Distinguishing between a torus and a toroid.
  • Understanding different types of tori like horn and spindle tori.
  • Calculating the surface area of a torus.
Torus vs. Toroid
While often used interchangeably, 'torus' specifically refers to the donut-shaped surface generated by a circle. A 'toroid' is a more general term for a surface generated by revolving any closed curve around an axis.
Surface Area of a Torus
Beyond volume, the surface area (A) of a torus is another important property, calculated with the formula A = (2πr)(2πR) = 4π²Rr. This can also be derived from Pappus's first theorem.
Non-Standard Tori
When the condition R > r is not met, different shapes emerge. A 'horn torus' (R=r) has no central hole, and a 'spindle torus' (R<r) self-intersects. These shapes have different volume and surface area formulas.