Ellipsoid Volume Calculator - Three-Axis Volume & Capacity Tool

An ellipsoid volume calculator is a geometry tool that finds the interior capacity of a three-axis ellipsoid, a smooth closed surface where every cross-section through one principal axis is an ellipse. The tool takes the three semi-axes a, b, and c (the distances from the center to the surface along the x, y, and z directions) and returns the volume with capacity conversions in liters, US gallons, and cubic feet.

• • Astronomy: Estimating interior volume of planets, moons, and asteroids modeled as oblate spheroids.
• • Engineering and tank design: Sizing pressure vessels, fuel tanks, and propellant containers shaped as ellipsoidal caps or full ellipsoids.
• • Biology and medical imaging: Estimating organ or tumor volumes from ellipsoid approximations of MRI or ultrasound measurements.
• • Education: Checking worked problems for geometry, calculus, and physics involving ellipsoidal integration.

In practice, almost every real 3D object deviates from a perfect sphere, and the ellipsoid is the next-simplest closed shape that captures that asymmetry. By giving the shape three independent axes, you can describe something squashed like a pumpkin or stretched like a rugby ball while keeping a single closed-form volume formula.

The three inputs to the calculator are the semi-axes, the half-widths of the ellipsoid along each direction. A sphere is the special case where a equals b equals c, with formula V = (4/3) * pi * r^3. Spheroids (two of three axes equal) are also handled automatically.

Once the volume is computed, the calculator converts the result into common capacity units. You can move from a geometry problem to a real-world number, like liters of fuel in a prolate tank, without reaching for a second tool.

Because a sphere is the limiting case of an ellipsoid where all three semi-axes are equal, our Sphere Volume Calculator returns the same result whenever a equals b equals c.

The calculator uses the standard closed-form ellipsoid volume formula and a chain of unit conversions to deliver the result in the unit system you need.

• a, b, c: The three semi-axes of the ellipsoid. Each is the distance from the center to the surface along one principal direction.
• pi (π): The mathematical constant 3.14159265358979..., the ratio of a circle's circumference to its diameter.
• Length unit: The shared length unit for all three axes. The volume output appears in the matching cubic unit before being converted to liters, US gallons, and cubic feet.

Geometrically, an ellipsoid is the result of scaling a unit sphere by a, b, and c along its three principal axes. The unit sphere has volume (4/3) * pi, and scaling each axis by k multiplies the volume by k, so the full product gives the ellipsoid volume (4/3) * pi * a * b * c.

If you choose the Diameter input mode, the calculator halves each input before applying the formula. That keeps the math correct regardless of whether you measured with calipers (diameter) or a coordinate measuring machine (semi-axis).

After the raw volume is computed, the calculator converts it into cubic meters using the cube of the unit-to-meter factor, then derives liters, US gallons, and cubic feet. The surface area uses Knud Thomsen's approximation, accurate to within about 1% for typical ellipsoid shapes.

According to Wolfram MathWorld, the volume of a triaxial ellipsoid is (4/3) * pi * a * b * c, where a, b, and c are the three semi-axes measured from the geometric center to the surface.

If you already have a volume in cubic meters from another tool, our Volume Converter lets you move between liters, US gallons, and cubic feet without re-entering the ellipsoid axes.

Before you measure an ellipsoid in the wild, it helps to be familiar with these four geometric ideas:

When you measure an ellipsoid, the largest source of error is usually confusion between a semi-axis and a diameter. If you grab a sphere with calipers and read 10 cm, that is the diameter, so the semi-axis to plug into the formula is 5 cm.

The other concept worth internalizing is that the formula is symmetric in a, b, and c. Swapping any two axes does not change the volume, so you do not need to worry about labeling. As long as the three numbers are correct, the result is correct.

When you need to compare an ellipsoidal tank to a straight-walled alternative, our Cylinder Volume Calculator gives the cylinder capacity for the same height and diameter.

The ellipsoid volume calculator workflow has five steps, and the result updates as you type:

1. 1 Choose an input mode: Pick Semi-Axes if you already have half-diameters from a coordinate measurement, or Diameters if you measured the full width with calipers or a tape.
2. 2 Enter the three axes: Type the three values into the Axis a, Axis b, and Axis c fields. Order does not matter.
3. 3 Select a length unit: Pick the unit that matches your measurements (cm, m, in, or ft). The primary volume output appears in the matching cubic unit.
4. 4 Read the volume and capacity: The total volume in the chosen cubic unit appears at the top, followed by liters, US gallons, and cubic feet.
5. 5 Note the surface area: The Knud Thomsen surface area approximation is shown last, useful for heat transfer, drag, and material-coating estimates.

For example, if a fuel tank is a prolate spheroid with a diameter of 1.2 m and length 2.4 m, choose Diameter mode, enter 1.2, 1.2, and 2.4 in meters, and read the volume (about 1.81 m³ or 1,810 L).

To see how a pointed cone stacks up against an ellipsoid of the same base diameter and height, our Cone Volume Calculator returns the cone volume for the same dimensions.

The tool saves time and reduces error across common workflows:

• • Handles all ellipsoid classes: Works for triaxial ellipsoids, oblate spheroids, prolate spheroids, and the sphere limit case.
• • Avoids manual unit math: Computes cubic unit, liters, US gallons, and cubic feet in a single step, so you do not chain separate conversion tools.
• • Accepts diameters or semi-axes: The input mode toggle removes the common mistake of forgetting to halve a measured diameter.
• • Real-time validation: Flags non-positive axes immediately, so you do not silently get a zero or negative volume.
• • Pairs volume with surface area: Shows the Thomsen surface area approximation alongside the volume, useful for thermal and material-coating problems.

If you need to compare an ellipsoid against a cylinder or a sphere, the liters and US gallons output makes the comparison direct. A prolate tank of 1.81 m³ holds roughly 478 US gallons, a number a procurement team can use without further conversion.

The volume output is also a starting point for mass and cost estimates. Multiply the cubic meter value by the density of the contents to get mass, or by a unit price to get a fill cost.

For a single page that handles several 3D shapes at once, our Volume Calculator is a useful complement to this ellipsoid volume calculator.

Several input and modeling choices change the final number:

• • The formula assumes a smooth, closed triaxial ellipsoid. Real objects with flat faces, dents, or holes will not match exactly.
• • The surface area is a numerical approximation, not an exact closed-form result. For high-precision surface area work, integrate numerically or use a dedicated CAD tool.

When the ellipsoid is very elongated or flattened, small input changes have a large effect on the volume because the formula is linear in each semi-axis. A 5% measurement error on the largest axis becomes a 5% volume error.

Astronomical models like the reference Earth ellipsoid WGS84 use polar and equatorial radii that differ by about 0.3%. That small difference is enough to shift Earth's implied volume by tens of millions of cubic kilometers, which is why geodesists prefer a fixed reference ellipsoid.

According to Wikipedia (Ellipsoid), an ellipsoid with two equal semi-axes is called a spheroid and is classified as oblate (flattened, like Earth) or prolate (elongated, like a rugby ball), while the general triaxial case has three independent semi-axes.