Ellipsoid Volume Calculator: Volume, Surface Area, and Equivalent Sphere
Enter the three semi-axes of your ellipsoid and this calculator returns the exact volume using V = (4/3) x pi x a x b x c, the approximate surface area via the Knud Thomsen formula, and the radius of the sphere that holds the same volume. Switch between metric and imperial units, and follow the step-by-step working panel to see how every number is reached.
Formula
Worked example
Semi-axes a = 5 cm, b = 3 cm, c = 4 cm. Product: 5 x 3 x 4 = 60. Volume = (4/3) x pi x 60 = 251.3274 cm³. Equivalent sphere radius = (3 x 251.3274 / (4 x pi))^(1/3) = 3.9149 cm. Surface area (Thomsen) = approximately 247.8 cm².
What is an ellipsoid?
An ellipsoid is a three-dimensional surface obtained by stretching or compressing a sphere along each of its three coordinate axes independently. Every cross-section cut through the center is an ellipse. The shape is defined by three values called semi-axes: a (half the length along the x-axis), b (half the length along the y-axis), and c (half the length along the z-axis).
When all three semi-axes are equal (a = b = c = r), the result is a perfect sphere. When exactly two are equal, the shape is called a spheroid: prolate if the unique axis is the longest (like a rugby ball or an American football), and oblate if the unique axis is the shortest (like Earth, which bulges at the equator). When all three differ, the shape is a triaxial ellipsoid, common in geology and biology.
The ellipsoid volume formula
The exact volume of an ellipsoid is:
V = (4/3) x pi x a x b x c
This is a direct extension of the sphere formula V = (4/3) x pi x r³, where r is replaced by the geometric mean of the three semi-axes. Because pi x a x b x c is the volume of the circumscribed rectangular box divided by (3 / (4/3)), the formula can also be read as roughly 52.36% of the bounding box volume (length x width x height).
In medical imaging, this 52.36% approximation is written as V = L x W x H x 0.5236, where L, W, and H are the full diameters (not semi-axes) of the organ being measured. That formula is mathematically identical to the general ellipsoid equation and is the standard method for estimating prostate, ovary, kidney, and bladder volumes from ultrasound or CT measurements.
Surface area and the Thomsen approximation
Unlike for spheres, there is no exact closed-form formula for the surface area of a general triaxial ellipsoid. The exact result requires an elliptic integral, which cannot be expressed in terms of elementary functions. Several approximations have been published; the most accurate simple one for practical ranges of axis ratios is the Knud Thomsen formula (2004):
S = 4 x pi x [(a^p x b^p + a^p x c^p + b^p x c^p) / 3]^(1/p), p = 1.6075
This approximation has a maximum relative error of about 1.06% for all axis ratios. For a sphere (a = b = c = r) it reduces exactly to the correct formula S = 4 x pi x r². For prolate and oblate spheroids, exact closed forms do exist (involving arcsin and arctanh respectively), but the Thomsen formula is close enough for engineering and scientific purposes and works for all three cases with one equation.
Medical imaging and the ellipsoid method
The ellipsoid volume formula is the international standard for estimating organ and lesion volumes from two-dimensional imaging. In ultrasound and MRI, a clinician measures the three perpendicular diameters of the structure (longest, anterior-posterior, and transverse dimensions), then applies V = L x W x H x 0.5236 to estimate volume without a 3-D reconstruction.
- Prostate volume: the standard method in urology, used to calculate PSA density (PSA level divided by prostate volume). A prostate volume above 30 mL is generally considered enlarged.
- Ovarian volume: used in gynecology to screen for ovarian masses. Normal adult ovaries are typically 6-10 mL each; volumes above 20 mL prompt further investigation.
- Kidney, spleen, and thyroid: similar ellipsoid estimates are applied routinely in abdominal and endocrine imaging.
- Tumor volumes: response to cancer treatment is tracked by comparing serial volume estimates, and the ellipsoid formula is used in the RECIST guideline as one standard measurement approach.
The 0.5236 correction factor (= pi/6) accounts for the fact that a real organ is not a rectangular box: the ellipsoid shape consistently occupies about 52% of the volume of its bounding box.
How to use this calculator
Enter the three semi-axes of your ellipsoid and choose your measurement unit. If you have full diameter measurements (from a ruler or an imaging report), divide each by two to get the semi-axis before entering it. The calculator returns the exact volume, the Thomsen surface area approximation, and the equivalent sphere radius instantly.
The chart below the results shows how the volume would change if you scaled each axis independently from 10% to 200% of its current value, keeping the other two fixed. This makes it easy to see which axis has the most leverage on total volume (doubling any axis doubles volume, but the three axes are fully independent).
The step-by-step panel shows every arithmetic operation with your exact numbers, so you can verify the result manually or copy the working into a report.
Common ellipsoid shapes and their formulas
| Shape | Condition | Formula | Example |
|---|---|---|---|
| Sphere | a = b = c = r | V = (4/3) pi r³ | Basketball, bubble |
| Prolate spheroid | a = b < c | V = (4/3) pi a² c | Rugby ball, lemon |
| Oblate spheroid | a = b > c | V = (4/3) pi a² c | Earth, M&M candy |
| Triaxial ellipsoid | a != b != c | V = (4/3) pi a b c | Potato, avocado |
| Medical approximation | L, W, H = diameters | V = L x W x H x 0.5236 | Organ volumes (radiology) |
Special cases of the general ellipsoid where two or more axes are equal.
Frequently asked questions
How do I calculate the volume of an ellipsoid?
Multiply (4/3) by pi (approximately 3.14159) and then by each of the three semi-axes: V = (4/3) x pi x a x b x c. For example, semi-axes of 5, 3, and 4 cm give V = 1.3333 x 3.14159 x 5 x 3 x 4 = 251.33 cm³. If you have the full diameters (L, W, H) rather than the semi-axes, use V = L x W x H x 0.5236 instead, which is the same formula rearranged.
What is the difference between an ellipsoid and a spheroid?
A spheroid is a special case of an ellipsoid where exactly two of the three semi-axes are equal. A prolate spheroid has two equal shorter axes and one longer axis (like a rugby ball), while an oblate spheroid has two equal longer axes and one shorter axis (like Earth, which is flattened at the poles). A general ellipsoid has all three axes different; a sphere is the further special case where all three axes are equal.
What is the medical ellipsoid formula and when is it used?
Radiologists and sonographers estimate organ volume using V = L x W x H x 0.5236, where L, W, and H are the full measured diameters in three perpendicular planes. This is mathematically identical to (4/3) x pi x a x b x c with a = L/2, b = W/2, c = H/2. It is the standard method for prostate, ovarian, kidney, and bladder volumes in ultrasound and CT imaging.
Can an ellipsoid have a larger volume than a sphere with the same longest axis?
Yes. If the sphere has radius equal to the longest semi-axis of the ellipsoid, then the sphere is larger (because the ellipsoid is compressed along the other two axes). But if the sphere has the same radius as the shortest semi-axis, the ellipsoid is larger. The ellipsoid volume equals the sphere volume only when the equivalent sphere radius equals the geometric mean of the three semi-axes, which this calculator computes for you.
Why is there no exact formula for the surface area of an ellipsoid?
The surface area of a general triaxial ellipsoid involves a Legendre elliptic integral, which has no closed form in elementary functions. Exact formulas do exist for the two spheroid special cases (using arcsin for prolate and arctanh for oblate), but for the general case you must either use a numerical integration or an approximation. The Thomsen formula used here is accurate to within about 1.1% for all realistic axis ratios and is the standard engineering choice.
How does the volume change if I double one semi-axis?
Because the volume formula is V = (4/3) x pi x a x b x c, it is directly proportional to each semi-axis independently. Doubling any single axis exactly doubles the volume. Doubling all three axes increases the volume by a factor of 8 (2 x 2 x 2), which matches the general rule that scaling all linear dimensions by a factor k multiplies volume by k³.
What is the equivalent sphere radius?
The equivalent sphere radius is the radius of a perfect sphere that has exactly the same volume as your ellipsoid. It equals the cube root of (3V / 4pi), or equivalently the geometric mean of the three semi-axes (a x b x c)^(1/3). It is useful when you need to compare volumes across objects of different shapes, or when a downstream formula assumes a spherical shape.