Input
Theory and Formulas
Consider a triaxial ellipsoid centered at the origin with semi-axes a, b, c aligned to the coordinate axes:
x^2/a^2 + y^2/b^2 + z^2/c^2 = 1The radius in a unit direction u = (ux, uy, uz) is
r(u) = 1 / sqrt( u_x^2/a^2 + u_y^2/b^2 + u_z^2/c^2 )The diameter (full length through the center) along that direction is
D(u) = 2 / sqrt( u_x^2/a^2 + u_y^2/b^2 + u_z^2/c^2 )- Body diagonal uses
u = (1,1,1)/√3, yieldingD = 2√3 / √(1/a^2 + 1/b^2 + 1/c^2). - Face diagonals use
u = (1,1,0)/√2, etc. - Axis diameters are simply
2a,2b,2c. - For comparison, the diagonal of the circumscribed cuboid is
2√(a^2 + b^2 + c^2).
Tip: For a sphere with a=b=c=R, every direction gives the same result D=2R.
Results
Enter values and choose a diagonal type, then press Calculate to see results with comparisons.
Examples
Frequently Asked Questions
1) What does “ellipsoid diagonal” mean here?
It is the full diameter measured through the center along a chosen direction vector. For boxes this corresponds to the space diagonal; for ellipsoids it depends on orientation.
2) How do I choose the correct diagonal type?
Use axis diameters for principal directions, face diagonals for diagonals within coordinate planes, the body diagonal for ⟨1,1,1⟩, or provide a custom vector for any other orientation.
3) Do I need to provide a unit?
Yes, choose one unit for all three semi-axes. The result uses the same unit, so be consistent when entering values.
4) What if my ellipsoid is rotated in space?
This tool assumes semi-axes are aligned with x, y, z. To handle a rotated ellipsoid, first transform your direction vector into the ellipsoid’s principal-axis frame.
5) Why is the requested diagonal sometimes larger than a principal diameter?
Depending on the relative sizes of a, b, c, some off-axis directions can produce diameters between the smallest and largest of 2a, 2b, 2c but never exceed the largest axis diameter.
6) Why compare with the circumscribed cuboid diagonal?
It provides an upper bound from the bounding box. The ellipsoid’s diagonal along any direction cannot exceed the cuboid’s space diagonal.
7) Can I download results?
Yes, use the CSV or JSON buttons in the Results card to export the current calculation for reporting or further analysis.
8) What precision should I use?
Use enough decimal places for your application. Engineering work often uses three to six decimals; scientific work may require more.