Torus Volume Calculator

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 seen in objects like inner tubes, rings, and some types of piping.

2. How Does the Calculator Work?

The calculator uses the torus volume formula:

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

Where:

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

Explanation: The formula accounts for both the size of the tube (minor radius) and how far it's rotated around the axis (major radius).

3. Importance of Torus Volume Calculation

Details: Calculating torus volume is important in engineering, architecture, and manufacturing where torus-shaped objects are used. It helps in material estimation, structural analysis, and design optimization.

4. Using the Calculator

Tips: Enter both radii in the same units (e.g., centimeters, inches). The major radius must be larger than the minor radius for a standard torus. All values must be positive numbers.

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 this formula be used for partial torus shapes?
A: No, this formula calculates the volume of a complete torus (360 degrees). Partial torus volumes require more complex calculations.

Q3: What if my major radius is smaller than minor radius?
A: This creates a self-intersecting torus (spindle torus). The formula still works mathematically but may not represent a physical object.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for an ideal torus shape. Real-world objects may have slight variations.

Q5: Can I use different units for R and r?
A: No, both radii must be in the same units to get a correct volume calculation.