🥚 Ellipsoid Volume Calculator
Calculate the volume of an ellipsoid from its three semi-axes
How to Use This Calculator
Enter Three Semi-Axes
Input the lengths of the three semi-axes (a, b, c) of the ellipsoid. These are half the lengths of the ellipsoid's three principal diameters.
Click Calculate
Press the "Calculate Volume" button to compute the ellipsoid volume using the formula V = (4/3)Ï€abc.
Review Result
View the calculated volume displayed in cubic units. The result includes the step-by-step calculation showing how the formula was applied.
Formula
V = (4/3) × π × a × b × c
Where a, b, and c are the three semi-axis lengths
Where:
- V = volume of the ellipsoid
- a = semi-axis length along the x-axis
- b = semi-axis length along the y-axis
- c = semi-axis length along the z-axis
- π ≈ 3.14159
Special Cases:
- Sphere: If a = b = c = r, then V = (4/3)πr³
- Prolate Spheroid: If a = b < c (rugby ball shape)
- Oblate Spheroid: If a = b > c (discus shape, like Earth)
Example 1: Find the volume of an ellipsoid with semi-axes a = 3, b = 4, c = 5 units
V = (4/3) × π × 3 × 4 × 5
V = (4/3) × π × 60
V = 80π ≈ 251.33 units³
Example 2: Find the volume of a sphere with radius 7 units (a = b = c = 7)
V = (4/3) × π × 7 × 7 × 7
V = (4/3) × π × 343
V = (1372/3)π ≈ 1,436.76 units³
Example 3: Earth is an oblate spheroid with semi-axes approximately 6,378 km (equatorial) and 6,357 km (polar)
a = b = 6,378 km, c = 6,357 km
V = (4/3) × π × 6,378 × 6,378 × 6,357
V ≈ 1.083 × 10¹² km³
About Ellipsoid Volume Calculator
An ellipsoid is a three-dimensional shape that generalizes a sphere. While a sphere has all radii equal, an ellipsoid has three different semi-axis lengths (a, b, c) along three perpendicular axes. This calculator computes the volume of any ellipsoid using the formula V = (4/3)Ï€abc.
When to Use This Calculator
- Geometry: Calculate volumes of ellipsoidal shapes in mathematical problems
- Earth Sciences: Estimate volumes of planets, moons, or other celestial bodies
- Engineering: Design ellipsoidal tanks, containers, or structures
- Medicine: Calculate volumes of ellipsoidal organs or structures from medical imaging
- Physics: Determine volume for ellipsoidal objects in fluid dynamics or mechanics
- Architecture: Plan volumes for ellipsoidal architectural elements
Why Use Our Calculator?
- ✅ Handles All Ellipsoids: Works for any ellipsoid shape (sphere, spheroid, or general ellipsoid)
- ✅ Instant Results: Get accurate volume calculations immediately
- ✅ Step-by-Step Display: See the formula applied with your values
- ✅ 100% Accurate: Uses precise mathematical formulas
- ✅ Educational: Helps understand ellipsoid geometry
- ✅ Completely Free: No registration required
Types of Ellipsoids
Depending on the relationship between the three semi-axes, ellipsoids take different forms:
- Sphere: a = b = c (all axes equal) - the most symmetric ellipsoid
- Prolate Spheroid: a = b < c (rugby ball, elongated along one axis)
- Oblate Spheroid: a = b > c (discus, flattened at the poles) - Earth is approximately this shape
- General Ellipsoid: a ≠b ≠c (three different axes) - most general case
Real-World Applications
Planetary Science: Earth is approximately an oblate spheroid with equatorial radius 6,378 km and polar radius 6,357 km. Its volume is about 1.083 × 10¹² km³, calculated using the ellipsoid volume formula.
Medical Imaging: Calculate the volume of organs from MRI or CT scans. If an organ has semi-axes of 5 cm, 6 cm, and 4 cm, its volume is (4/3) × π × 5 × 6 × 4 ≈ 502.65 cm³.
Engineering: Design an ellipsoidal tank with semi-axes 2 m, 2 m, and 3 m. The volume is (4/3) × π × 2 × 2 × 3 = 16π ≈ 50.27 m³, useful for capacity planning.
Frequently Asked Questions
What is an ellipsoid?
An ellipsoid is a 3D shape that generalizes a sphere. While a sphere has equal radii in all directions, an ellipsoid has three different semi-axis lengths (a, b, c) along three perpendicular axes, creating an egg-like or rugby ball-like shape.
How is an ellipsoid different from a sphere?
A sphere is a special case of an ellipsoid where all three semi-axes are equal (a = b = c). An ellipsoid allows the three axes to be different, creating more varied shapes. The sphere volume formula V = (4/3)πr³ is a special case of the ellipsoid formula when a = b = c = r.
What are semi-axes?
Semi-axes are half the lengths of the ellipsoid's three principal diameters. If the full lengths along x, y, and z axes are 2a, 2b, and 2c, then a, b, and c are the semi-axes. They extend from the center to the surface.
Can I use this for a sphere?
Yes! If you enter the same value for all three semi-axes (a = b = c = radius), you'll get the sphere volume. For example, for a sphere of radius 5, enter a = 5, b = 5, c = 5.
What's the difference between a prolate and oblate spheroid?
A prolate spheroid has a = b < c (elongated like a rugby ball). An oblate spheroid has a = b > c (flattened like a discus). Earth is an oblate spheroid, slightly flattened at the poles.
How accurate is this calculator?
This calculator uses the exact mathematical formula V = (4/3)Ï€abc, providing precise results limited only by input accuracy and floating-point precision. For most practical purposes, results are accurate to many decimal places.