Calculator Input
Formula Used
If a = b = c = r, then:
S = 4πr2
If two semi-axes are equal and the polar axis is smaller, use:
S = 2πr2 [1 + ((1 - e2)/e) atanh(e)]
where e = √(1 - p2/r2)
If two semi-axes are equal and the polar axis is larger, use:
S = 2πr2 [1 + (p/(re)) sin-1(e)]
where e = √(1 - r2/p2)
When all three semi-axes differ, this tool uses Knud Thomsen's engineering approximation:
S ≈ 4π[(apbp + apcp + bpcp)/3]1/p
with p = 1.6075
This mixed approach keeps the calculator practical, fast, and accurate for engineering work. Exact formulas are used for spheres and spheroids. The triaxial result uses a trusted approximation because a simple elementary closed form is not available.
How to Use This Calculator
- Enter the three semi-axis lengths a, b, and c.
- Choose the measurement unit you want displayed.
- Select the decimal precision for the final result.
- Keep Auto mode for exact spheroid handling.
- Press the calculate button to generate results.
- Review the surface area, shape type, and method.
- Inspect the Plotly graph to verify geometry.
- Export the summary as CSV or PDF.
Example Data Table
| Case | a | b | c | Type | Method | Surface Area (m2) |
|---|---|---|---|---|---|---|
| Example 1 | 4 | 4 | 4 | Sphere | Exact | 201.062 |
| Example 2 | 5 | 5 | 3 | Oblate spheroid | Exact | 234.736 |
| Example 3 | 3 | 3 | 6 | Prolate spheroid | Exact | 193.306 |
| Example 4 | 6 | 4 | 2 | Triaxial ellipsoid | Approximation | 195.888 |
Frequently Asked Questions
1) What does this calculator measure?
It estimates the outer surface area of an ellipsoid using your three semi-axis lengths. It also identifies the ellipsoid type and shows the calculation method used.
2) What are semi-axes a, b, and c?
They are half-lengths measured from the ellipsoid center to its surface along three perpendicular directions. Enter consistent units for all three values.
3) Is the triaxial result exact?
No. A general triaxial ellipsoid lacks a simple elementary surface formula. This calculator uses Knud Thomsen's approximation, which is widely accepted for practical engineering work.
4) When does the calculator use exact equations?
Auto mode uses exact equations for spheres, oblate spheroids, and prolate spheroids. Those cases happen when all three axes match, or when exactly two axes match.
5) Why is the result shown in squared units?
Surface area measures coverage over a surface, not length. That is why meters become square meters, and centimeters become square centimeters.
6) Can I use inches, feet, or millimeters?
Yes. The calculator is unit-flexible. Enter all three semi-axes in the same unit, and the final surface area will be shown in that unit squared.
7) What does eccentricity mean here?
Eccentricity describes how stretched or flattened a spheroid is. A value near zero is closer to a sphere. Larger values indicate stronger deformation.
8) Why is the 3D graph useful?
The graph helps you visually confirm whether your dimensions describe a sphere, flattened spheroid, elongated spheroid, or a fully triaxial ellipsoid.