Calculate Volume of a Torus

Torus Volume Formula:

\[ V = 2 \pi^2 R r^2 \]

Volume (V):

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1. What is a Torus?

A torus is a doughnut-shaped surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. It's commonly found in physics, engineering, and architecture.

2. How Does the Calculator Work?

The calculator uses the torus volume formula:

\[ V = 2 \pi^2 R r^2 \]

Where:

\( V \) — Volume (cubic length units)
\( \pi \) — Mathematical constant (~3.14159)
\( R \) — Major radius (distance from center of tube to center of torus)
\( r \) — Minor radius (radius of the tube)

Explanation: The formula calculates the volume by multiplying the area of the cross-section (πr²) by the circumference of the path traced by the center of the cross-section (2πR).

3. Importance of Torus Volume Calculation

Details: Calculating torus volume is essential in various fields like mechanical engineering (for designing pipes and rings), physics (for modeling magnetic fields), and architecture (for creating torus-shaped structures).

4. Using the Calculator

Tips: Enter both major and minor radii in the same length units. Both values must be positive numbers. The result will be in cubic units of whatever length unit you used.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between major and minor radius?
A: The major radius (R) is the distance from the center of the torus to the center of the tube, while the minor radius (r) is the radius of the tube itself.

Q2: Can I use diameter instead of radius?
A: Yes, but remember to divide diameters by 2 to get the radius values needed for the formula.

Q3: What are some real-world examples of tori?
A: Common examples include doughnuts, lifebuoys, O-rings, and the shape of some fusion reactors (tokamaks).

Q4: How does this relate to surface area?
A: The surface area of a torus is different from its volume. The surface area formula is \( A = 4\pi^2 R r \).

Q5: What if the minor radius is larger than the major radius?
A: This creates a self-intersecting torus (a spindle torus), and this calculator's formula doesn't apply to that case.