Ellipsoid — Definition, Formula & Examples

Q: What is the difference between an ellipsoid and a sphere?

A sphere is a special case of an ellipsoid where all three semi-axes are equal (a = b = c = r). In a general ellipsoid, the three semi-axes can have different lengths, so the shape is stretched or compressed in one or more directions. The sphere equation x² + y² + z² = r² is what you get when you set a = b = c in the ellipsoid equation.

Q: What is the difference between an ellipsoid and a spheroid?

A spheroid is an ellipsoid that has two equal semi-axes. If the two equal axes are longer than the third, it is an oblate spheroid (like Earth, slightly flattened at the poles). If the two equal axes are shorter, it is a prolate spheroid (like a rugby ball). A general ellipsoid has three distinct semi-axis lengths.

Q: How do you find the surface area of an ellipsoid?

There is no simple closed-form formula for the surface area of a general ellipsoid — it requires elliptic integrals. However, an approximation is given by Knud Thomsen's formula: S ≈ 4π[(aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3]^(1/p) where p ≈ 1.6075. For a sphere (a = b = c = r), this correctly reduces to 4πr².

Ellipsoid

A sphere-like surface for which all cross-sections are ellipses.

3D ellipsoid shape with three perpendicular dashed axes shown, resembling a sphere but with elliptical cross-sections.

See also

Oblate spheroid, prolate spheroid

Key Formula

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

Where:

$x, y, z$ = Coordinates of any point on the surface 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

Worked Example

Problem: Determine whether the point (1, 2, 2) lies on, inside, or outside the ellipsoid with semi-axes a = 2, b = 4, and c = 3.

Step 1: Write the standard equation of the ellipsoid with the given semi-axes.

\frac{x^2}{2^2} + \frac{y^2}{4^2} + \frac{z^2}{3^2} = 1 \quad\Longrightarrow\quad \frac{x^2}{4} + \frac{y^2}{16} + \frac{z^2}{9} = 1

Step 2: Substitute the point (1, 2, 2) into the left-hand side of the equation.

\frac{1^2}{4} + \frac{2^2}{16} + \frac{2^2}{9} = \frac{1}{4} + \frac{4}{16} + \frac{4}{9}

Step 3: Simplify each fraction and find a common denominator (144).

\frac{1}{4} = \frac{36}{144}, \quad \frac{4}{16} = \frac{1}{4} = \frac{36}{144}, \quad \frac{4}{9} = \frac{64}{144}

Step 4: Add the three fractions.

\frac{36}{144} + \frac{36}{144} + \frac{64}{144} = \frac{136}{144} = \frac{17}{18} \approx 0.944

Step 5: Compare the result to 1. Since 17/18 < 1, the point lies inside the ellipsoid.

\frac{17}{18} < 1 \quad\Longrightarrow\quad \text{inside}

Answer: The point (1, 2, 2) lies inside the ellipsoid because the sum equals 17/18, which is less than 1.

Another Example

This example uses the volume formula rather than the surface equation, showing a common applied calculation students need for ellipsoids.

Problem: Find the volume of an ellipsoid with semi-axes a = 3, b = 4, and c = 5.

Step 1: Recall the volume formula for an ellipsoid.

V = \frac{4}{3}\pi\, a\, b\, c

Step 2: Substitute the given semi-axis lengths.

V = \frac{4}{3}\pi (3)(4)(5)

Step 3: Multiply the semi-axes together: 3 × 4 × 5 = 60.

V = \frac{4}{3}\pi \times 60 = \frac{240}{3}\pi = 80\pi

Step 4: Compute the decimal approximation.

V \approx 80 \times 3.1416 \approx 251.33

Answer: The volume of the ellipsoid is 80π ≈ 251.33 cubic units.

Frequently Asked Questions

What is the difference between an ellipsoid and a sphere?

A sphere is a special case of an ellipsoid where all three semi-axes are equal (a = b = c = r). In a general ellipsoid, the three semi-axes can have different lengths, so the shape is stretched or compressed in one or more directions. The sphere equation x² + y² + z² = r² is what you get when you set a = b = c in the ellipsoid equation.

What is the difference between an ellipsoid and a spheroid?

A spheroid is an ellipsoid that has two equal semi-axes. If the two equal axes are longer than the third, it is an oblate spheroid (like Earth, slightly flattened at the poles). If the two equal axes are shorter, it is a prolate spheroid (like a rugby ball). A general ellipsoid has three distinct semi-axis lengths.

How do you find the surface area of an ellipsoid?

There is no simple closed-form formula for the surface area of a general ellipsoid — it requires elliptic integrals. However, an approximation is given by Knud Thomsen's formula: S ≈ 4π[(aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3]^(1/p) where p ≈ 1.6075. For a sphere (a = b = c = r), this correctly reduces to 4πr².

Ellipsoid vs. Sphere

	Ellipsoid	Sphere
Definition	3D surface with three semi-axes a, b, c (possibly all different)	3D surface where every point is the same distance r from the center
Standard equation	x²/a² + y²/b² + z²/c² = 1	x² + y² + z² = r²
Volume	(4/3)πabc	(4/3)πr³
Cross-sections	Ellipses (or circles as special cases)	Always circles
Number of parameters	Three (a, b, c)	One (r)

Why It Matters

Ellipsoids appear across science and engineering. Earth is modeled as an oblate spheroid (a type of ellipsoid) in geodesy and GPS calculations, where assuming a perfect sphere would introduce significant errors. In physics, the moment of inertia of an ellipsoidal solid is a standard calculation, and in statistics, confidence regions for multivariate data are ellipsoidal in shape.

Common Mistakes

Mistake: Confusing semi-axes with full axes. Students sometimes plug the full length of an axis (e.g., 10) into the formula where the semi-axis (half-length, e.g., 5) is required.

Correction: The values a, b, and c in the standard equation are always half the total length along each axis. If the ellipsoid extends from −5 to 5 along the x-axis, then a = 5, not 10.

Mistake: Assuming that any cross-section of an ellipsoid is a circle. Students sometimes forget that circular cross-sections only occur in special planes.

Correction: In a general ellipsoid with three distinct semi-axes, most cross-sections are ellipses. Circular cross-sections exist only along specific planes. Only when at least two semi-axes are equal (spheroid) do you get a family of circular cross-sections.

Related Terms

Sphere — Special ellipsoid with all semi-axes equal
Surface — General category that includes ellipsoids
Ellipse — 2D cross-section shape of an ellipsoid
Oblate Spheroid — Ellipsoid with two equal longer semi-axes
Prolate Spheroid — Ellipsoid with two equal shorter semi-axes
Quadric Surface — Family of second-degree surfaces including ellipsoids
Volume — Key measurement computed for ellipsoids