Ellipsoid Volume Calculator
Enter the three semi-axes of an ellipsoid — a stretched or squashed sphere — for its volume and approximate surface area, drawn to scale. Click any axis on the drawing to change it.
Dimensions
EllipsoidResult
Volume of an ellipsoid
An ellipsoid is a sphere scaled differently along three perpendicular axes. Its volume generalises the sphere directly, V = 4⁄3·πabc. The surface area has no exact elementary form, so a close approximation is used.
Editing on the drawing
Each axis is editable — click an axis length on the figure, type a new value and press Enter. The figure shows the three central cross-sections.
How to find the volume of an ellipsoid
An ellipsoid is a sphere stretched or squashed by a different amount along each of three perpendicular directions. Three measurements — the semi-axes a, b and c — describe it. When all three are equal it is simply a sphere.
The formulas
Volume follows the sphere's rule with the three semi-axes in place of one radius. Surface area, by contrast, has no exact elementary formula; this calculator uses the Knud Thomsen approximation, accurate to roughly one percent.
Worked example
Take an ellipsoid with semi-axes 56, 40 and 30.
Four-thirds of π times the three semi-axes.
Thomsen approximation (about 1% error).
What else the calculator returns
You also get the three full axis lengths (2a, 2b, 2c) — the overall width, depth and height. The drawing shows the three central ellipses, which read clearly as an ellipsoid without needing a perfect silhouette.
Where it comes up
Ellipsoid volumes describe rugby and AFL balls, planets and their equatorial bulge, lenses, pressure vessels and a great deal of scientific modelling.
Frequently asked questions
How do you find the volume of an ellipsoid?
Multiply 4⁄3π by the three semi-axes: V = 4⁄3·πabc.
Why is the surface area only approximate?
Unlike the volume, an ellipsoid's surface area has no closed elementary formula — it needs elliptic integrals. The Thomsen approximation used here is within about one percent.
What is the difference between a semi-axis and an axis?
A semi-axis runs from the centre to the surface, like a radius. The full axis is twice that, measured edge to edge.
When does an ellipsoid become a sphere?
When all three semi-axes are equal (a = b = c), the formulas reduce to 4⁄3πr³ and 4πr².
What is a prolate or oblate spheroid?
An ellipsoid with two equal axes. Prolate is egg-shaped (stretched along one axis); oblate is flattened, like a planet.