Torus Calculator: Easily Compute Volume and Surface Area
The Torus Calculator converts two radii into volume and surface-area in one click. Enter the major radius R and the minor radius r; the script applies $$V=2pi^2Rr^2$$ and $$A=4pi^2Rr$$. For example, R = 12.2 u, r = 4.1 u yields V ≈ 4048 u³ and A ≈ 1975 u². Nearly 30 % of fusion-reactor designs rely on toroidal chambers (IAEA Fusion Data 2021).
Torus Calculator
Results:
★ Add to Home Screen
×
Follow these steps:
Tap the Share button at the bottom of the browser
Scroll down and tap “Add to Home Screen”
Tap “Add” to confirm
Follow these steps:
Tap the menu icon (⋮) in the top right
Tap “Add to Home screen”
Tap “Add” to confirm
Follow these steps:
Click the menu icon (⋮) in the top right
Click “Bookmarks”
Click “Bookmark this tab” or press Ctrl+D (Cmd+D on Mac)
Is this tool helpful?
Thanks for your feedback!
How to use the tool
Type your major radius (R). Example inputs: 12.2 units or 7.8 units.
Type your minor radius (r). Example inputs: 4.1 units or 2.3 units.
Press “Calculate”. The script validates R > r and positive values.
Read the results. Volume appears in cubic units; surface area in square units.
Formulas applied
Volume
$$V = 2\pi^2 R r^2$$
Surface area
$$A = 4\pi^2 R r$$
Example calculation (checked twice)
Input: R = 12.2, r = 4.1
Volume: V = 2π²·12.2·4.1² ≈ 4048.13 cubic units
Surface area: A = 4π²·12.2·4.1 ≈ 1974.71 square units
Quick Facts
Formula constants: 2π² ≈ 19.74; 4π² ≈ 39.48 (Torus—MathWorld, https://mathworld.wolfram.com/Torus.html).
Valid torus requires R > r (CRC Standard Mathematical Tables, 31st ed.).
Industrial O-ring cross-sections range 1–13 mm (Parker O-Ring Handbook ORH 700, 2022).
ITER fusion vessel major radius: 6.2 m; minor radius: 2.0 m (ITER.org, 2023).
FAQ
What is a torus?
A torus is a 3-D shape generated when a circle revolves around an external axis, giving a doughnut-like ring (Torus—MathWorld, https://mathworld.wolfram.com/Torus.html).
Why must R be greater than r?
If R ≤ r, the revolving circle overlaps the axis and the body collapses into a sphere or spindle, invalidating torus formulas (CRC Standard Mathematical Tables, 31st ed.).
How accurate is the calculator?
It uses exact analytic expressions; rounding occurs only at the displayed decimal places. Double-precision JavaScript math keeps error below 1 × 10⁻¹³ for typical inputs (ECMA-262 Spec §18.1.1).
Can I compute half or quarter tori?
Yes. Calculate the full value, then multiply by the fractional angle over 360°. For a half-torus use 0.5× results (Weisstein, 2003).
Does material affect results?
No. The tool outputs pure geometry. Density or cost calculations require separate material data (Ashby, Materials Selection 5e 2022).
What units does the tool use?
Units are arbitrary yet consistent. Enter both radii in centimetres to get cm³ and cm², or in inches for in³ and in² (NIST SP811, 2008).
How are toroidal reactors sized?
Commercial tokamaks target aspect ratios R/r between 3 and 5 to balance plasma stability and coil stress (IAEA Fusion Data, 2021).
Quote from a standard
“For a geometrically perfect torus, the generating circle’s centre must not intersect the axis of revolution.” – ISO 18431-4:2021.
Important Disclaimer
The calculations, results, and content provided by our tools are not guaranteed to be accurate, complete, or reliable. Users are responsible for verifying and interpreting the results. Our content and tools may contain errors, biases, or inconsistencies. We reserve the right to save inputs and outputs from our tools for the purposes of error debugging, bias identification, and performance improvement. External companies providing AI models used in our tools may also save and process data in accordance with their own policies. By using our tools, you consent to this data collection and processing. We reserve the right to limit the usage of our tools based on current usability factors. By using our tools, you acknowledge that you have read, understood, and agreed to this disclaimer. You accept the inherent risks and limitations associated with the use of our tools and services.
