What is an ellipsoid?

A 3D shape like a stretched or squashed sphere. Defined by three semi-axes a, b, c. Sphere when a = b = c.

What is the volume of an ellipsoid?

V = (4/3)πabc. Exact formula; order of a, b, c does not matter.

Is there an exact formula for ellipsoid surface area?

No simple closed form for general a≠b≠c. The Knud Thomsen approximation is used (~1% accuracy).

Ψ = π^(1/3)(6V)^(2/3)/A. Equals 1 for a sphere; less than 1 for ellipsoids. Measures how round the shape is.

What is an oblate vs prolate spheroid?

Oblate: a = b > c (flattened, like Earth). Prolate: a > b = c (elongated, like a rugby ball).

Where are ellipsoids used?

Earth model (geodesy), eggs, rugby balls, tumors in medical imaging, and many natural shapes.

5 more

GEOMETRY3D GeometryMathematics Calculator

🥚

Ellipsoid — Stretched Sphere

Compute volume, surface area (Knud Thomsen approximation), and sphericity from three semi-axes. The ellipsoid generalizes the sphere—Earth, eggs, and rugby balls are ellipsoids.

Concept Fundamentals

V = (4/3)πabc

Volume

Knud Thomsen approx

Surface Area

Ψ = π^(1/3)(6V)^(2/3)/A

Sphericity

a=b=c → sphere

Sphere Case

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Volume is symmetric in a, b, c—order does not matter Sphericity Ψ = 1 only for a perfect sphere; ellipsoids have Ψ < 1 Earth is an oblate spheroid (a=b>c); rugby ball is prolate (a>b=c) Exact ellipsoid surface area requires elliptic integrals Knud Thomsen approximation is accurate to ~1.061%

Key quantities

V = (4/3)πabc

Volume

Key relation

Knud Thomsen approx

Surface Area

Key relation

Ψ = π^(1/3)(6V)^(2/3)/A

Sphericity

Key relation

a=b=c → sphere

Sphere Case

Key relation

Ready to run the numbers?

Why: Ellipsoids model Earth, eggs, rugby balls, tumors in medical imaging, and many natural shapes. Understanding ellipsoid geometry is essential for geodesy, packaging, and materials science.

How: Enter the three semi-axes (half-lengths of the principal axes). Volume uses the exact formula (4/3)πabc. Surface area uses the Knud Thomsen approximation (~1% accuracy) since no closed-form exists for general ellipsoids. Sphericity measures how round the shape is.

Volume is symmetric in a, b, c—order does not matterSphericity Ψ = 1 only for a perfect sphere; ellipsoids have Ψ < 1

Sources:Wolfram MathWorld — EllipsoidKhan Academy — 3D Geometry

Run the calculator when you are ready.

Ellipsoid CalculatorEnter semi-axes a, b, c to compute volume, surface area, and sphericity

🥚

3D GEOMETRYEllipsoid

Ellipsoid — Stretched Sphere

Volume V=(4/3)πabc. Surface area via Knud Thomsen. Sphericity measures roundness.

🥚 Sample Examples — Click to Load

Semi-Axes (a, b, c)

Semi-axis a

Semi-axis b

Semi-axis c

ellipsoid_volume.sh

CALCULATED

$ calc --a=7 --b=5 --c=3

Volume

439.823

Surface Area

308.4614

Sphericity

0.9067

Ellipsoid Properties

V = 439.82

SA ≈ 308.46 | Ψ = 0.9067

numbervibe.com/calculators/mathematics/3d-geometry/ellipsoid-volume-calculator

3D Visualization

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

INPUT

Semi-axes a, b, c

7, 5, 3

VOLUME

Volume V

439.823

V = (4/3)\text{pi} ext{abc}

SURFACE

Surface Area (approx)

308.4614

ext{Knud} ext{Thomsen} approximation

SHAPE

Sphericity Ψ

0.9067

Ψ = \text{pi} ^(1/3)(6V)^(2/3)/A

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🌍

Earth is an oblate spheroid—equatorial radius ~21 km larger than polar

— Geodesy

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Eggs are prolate spheroids—elongated along one axis

— Biology

🏉

Rugby balls approximate prolate spheroids for aerodynamic flight

— Sports Engineering

🔬

Tumors in medical imaging are often modeled as ellipsoids for volume estimation

— Radiology

📋 Key Takeaways

• Ellipsoid volume: $V = \\frac{4}{3}\\pi abc$ — sphere when a=b=c
• Surface area uses Knud Thomsen approximation (no closed-form for general ellipsoid)
• Sphericity Ψ = 1 for a sphere; <1 for elongated or flattened shapes
• Earth is an oblate spheroid (a=b>c); rugby ball is prolate (a>b=c)

💡 Did You Know?

🌍Earth is an oblate spheroid — equatorial radius ~21 km larger than polar

🥚Eggs are prolate spheroids — elongated along one axis

🏉Rugby balls approximate prolate spheroids for aerodynamic flight

🔬Tumors in medical imaging are often modeled as ellipsoids for volume estimation

📐Exact ellipsoid surface area requires elliptic integrals — no simple formula

💎Many gemstones are cut into ellipsoidal (oval) shapes

🪨Pebbles and river stones often have ellipsoid-like shapes from erosion

📖 How It Works

Volume

Volume = (4/3)πabc. Same structure as sphere volume $\\frac{4}{3}\\pi r^3$ when a=b=c=r.

Surface Area

No elementary formula exists. We use the Knud Thomsen approximation, accurate to ~1.061%.

Sphericity

Measures how spherical a shape is. Ψ=1 for sphere; lower values indicate elongation or flattening.

🎯 Expert Tips

Unit Consistency

Use same units for a, b, c. Volume in cubic units, surface area in square units.

Order Independence

Volume is unchanged if you swap a, b, c — multiplication is commutative.

Special Cases

a=b=c → sphere. a=b≠c → spheroid (oblate or prolate).

Real Measurements

Measure longest, middle, shortest semi-axes for best accuracy.

⚖️ Comparison

Shape	Volume Formula
Sphere	(4/3)πr³
Ellipsoid	(4/3)πabc
Cylinder	πr²h

❓ FAQ

What are semi-axes?

Half-lengths of the principal axes from center to surface. a, b, c are the three perpendicular radii.

Does order of a,b,c matter?

No. Volume = (4/3)πabc is symmetric in a, b, c.

Can I get exact surface area?

Exact formula requires elliptic integrals. Knud Thomsen gives ~1% accuracy.

Ellipsoid vs ellipse?

Ellipse is 2D; ellipsoid is its 3D analogue (like sphere to circle).

When is sphericity 1?

Only for a perfect sphere. All ellipsoids have Ψ < 1.

Earth's shape?

Earth is an oblate spheroid: a≈b≈6378 km, c≈6357 km (flattened at poles).

📊 Stats

Semi-axes

Volume formula

~1%

SA accuracy

Ψ≤1

Sphericity

📚 Sources

Wolfram MathWorld — Ellipsoid ↗

Wikipedia — Ellipsoid ↗

Knud Thomsen Formula ↗

Khan Academy — 3D Geometry ↗

⚠️ Disclaimer: Surface area uses Knud Thomsen approximation. For critical applications verify with specialized tools.

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⬅️Jump in and explore the concept!

Ellipsoid — Stretched Sphere

Ellipsoid — Stretched Sphere

🥚 Sample Examples — Click to Load

Semi-Axes (a, b, c)

3D Visualization

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How It Works

Volume

Surface Area

Sphericity

🎯 Expert Tips

Unit Consistency

Order Independence

Special Cases

Real Measurements

⚖️ Comparison

❓ FAQ

What are semi-axes?

Does order of a,b,c matter?

Can I get exact surface area?

Ellipsoid vs ellipse?

When is sphericity 1?

Earth's shape?

📊 Stats

📚 Sources

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Ellipsoid — Stretched Sphere

Ellipsoid — Stretched Sphere

🥚 Sample Examples — Click to Load

Semi-Axes (a, b, c)

3D Visualization

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How It Works

Volume

Surface Area

Sphericity

🎯 Expert Tips

Unit Consistency

Order Independence

Special Cases

Real Measurements

⚖️ Comparison

❓ FAQ

What are semi-axes?

Does order of a,b,c matter?

Can I get exact surface area?

Ellipsoid vs ellipse?

When is sphericity 1?

Earth's shape?

📊 Stats

📚 Sources

Related Mathematics Calculators

Related Calculators

Pyramid Volume Calculator

Tetrahedron Volume Calculator

Torus Volume Calculator

Cube Calculator

Sphere Calculator

Cylinder Calculator

We Value Your Privacy