Ellipsoid Volume Calculator
Compute the exact volume and approximate surface area of a triaxial ellipsoid from its three semi-axes a, b and c.
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V = (4/3)·π·a·b·c
25.13
Volume (4/3·π·a·b·c)- 1
Product of the semi-axes
3 × 2 × 1 = 6 - 2
Volume = (4 ÷ 3) × π × a × b × c
(4 ÷ 3) × π × 6 = 25.13Generalises the sphere formula (4/3)πr³ to three distinct radii.
How does this calculator work?
An ellipsoid with semi-axes a, b, c has exact volume V = (4/3)·π·a·b·c. Its surface area has no simple closed form, so we use the Thomsen approximation S ≈ 4π·((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p = 1.6075, accurate to about 1%. Enter the three positive semi-axes to get both results.
Formula
How this is calculated
An ellipsoid is the 3D generalization of an ellipse, defined by three mutually perpendicular semi-axes a, b and c (the half-lengths along the x, y and z directions). Enter each semi-axis in any consistent length unit; all three must be positive numbers.
The volume has an exact closed form, V = (4/3)·π·a·b·c. When all three semi-axes are equal this reduces to the familiar sphere volume (4/3)πr³, and when two are equal it gives a spheroid. Volume is returned in cubic units of whatever length unit you used.
The surface area of a general triaxial ellipsoid has no simple exact formula (it requires elliptic integrals), so we use the Thomsen / Knud Thomsen approximation S ≈ 4π·((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p = 1.6075. This is accurate to within about 1.06% for any ellipsoid and is exact for a sphere. Surface area is reported in square units. Inputs that are zero or negative are rejected since they do not describe a valid solid.
Frequently asked questions
Yes. V = (4/3)·π·a·b·c is the exact volume of any ellipsoid, generalizing the sphere formula (4/3)πr³.
A triaxial ellipsoid surface area requires elliptic integrals with no elementary closed form. The Thomsen approximation (p = 1.6075) used here is within about 1.06% for all ellipsoids and exact for a sphere.
A spheroid is an ellipsoid with two equal semi-axes (e.g. a = b ≠ c). Set two of the inputs equal to model a prolate or oblate spheroid; set all three equal for a sphere.
Also known as
TG we-Calculate Editorial Team. (2026). Ellipsoid Volume Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/ellipsoid-volume-calculator
TG we-Calculate Editorial Team. "Ellipsoid Volume Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/ellipsoid-volume-calculator.
TG we-Calculate Editorial Team, "Ellipsoid Volume Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/ellipsoid-volume-calculator
@misc{wecalculate_ellipsoid_volume_calculator, title = {Ellipsoid Volume Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/ellipsoid-volume-calculator}}, year = {2026}, note = {TG we-Calculate} }
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