What Is the Torus Surface Area and Volume Calculator?
A torus is a 3D surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
Common real-world physical examples include O-rings, toroidal electromagnetic inductors, inner tubes, and magnetic tokamak fusion reactors.
How Does the Torus Surface Area and Volume Calculator Work?
Pappus’s Centroid Theorem states that the volume generated equals the tube cross-sectional area (π·r²) multiplied by the distance traveled by its centroid (2π·R).
The surface area equals the circle’s perimeter (2π·r) multiplied by the centroid path length (2π·R), yielding 4π²·R·r.
Torus Surface Area and Volume Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
Torus volume equals 2π²Rr² and total surface area equals 4π²Rr, according to Pappus’s Centroid Theorem.
How to Use the Torus Surface Area and Volume Calculator
- Enter the major radius R (center of torus to tube center).
- Enter the minor radius r (radius of the circular tube).
- Review torus volume, surface area, tube cross-sectional area, and inner aperture dimensions.
Step-by-Step Example Calculation
Ring Torus (R = 5, r = 2)
Input Values:
Understanding Your Result
When R > r, a standard ring torus exists with an open center hole.
When R = r, a horn torus is formed where the center aperture touches a single point.
Factors That Affect the Result
- Aspect Ratio: Large R/r ratios produce slender thin rings, while R/r close to 1 yields fat doughy shapes.
- Scaling: Doubling both radii increases surface area by 4× and volume by 8×.
When Should You Use This Calculator?
- Designing rubber O-ring seals, gaskets, and mechanical fluid packing.
- Calculating toroidal core transformer coil winding surface areas and containment volumes.
Assumptions & Limitations
- Assumes a regular circular ring torus where r ≤ R.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Uses high-precision double arithmetic for transcendental constant π.