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Math calculator

Ellipsoid Calculator

Ellipsoid V = 4/3 πabc from three semi-axes. a = b = c is a sphere. Same engine as Solid Geometry. Not /sphere. Not the 2-D ellipse. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 66/66 tests · Production surface contract 3/3 · v1.11.0
Input interpretation
Enter values to calculate.
Result
—
Model
Triaxial ellipsoid volume from three semi-axes. Sphere surface when a=b=c.
Scope
Euclidean 3-space
Verification
Engine tested · 66/66 tests · Production surface contract 3/3 · Specification checked · v1.11.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
25 golden · 13 boundary · 28 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.11.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

VolumeV = 4/3 πabc
Sphere identitya = b = c ⇒ V = 4/3 πa³, A = 4πa²
iSame engine as /solid-geometry. a=b=c matches /sphere. The 2-D ellipse stays /ellipse. Triaxial surface (elliptic integrals) comes later. Not a second engine.

How to use

1

Enter a, b, and c

All three semi-axes must be positive. Default a = 2, b = 1.5, c = 1.

2

Read V

2-1.5-1 has V = 4π. Equal axes a=b=c=1 match the unit sphere.

Example calculations

Common configurations with formula and result.

ϟ

Triaxial 2-1.5-1

a = 2, b = 1.5, c = 1

V = 4π
4π
ϟ

Sphere identity

a = 1, b = 1, c = 1

V = 4π/3
4π/3

Ellipsoid calculator specification

Version 1.11.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A triaxial ellipsoid with semi-axes a>0, b>0, c>0 has volume 4/3 πabc. When a=b=c the solid is a sphere and this page reports the sphere surface 4πa². This page is mode=ellipsoid on math.geometry.solid. Not /sphere. Not the 2-D ellipse. Triaxial surface later. Not CAS.
What it calculates
Triaxial ellipsoid volume from three semi-axes. Sphere surface when a=b=c.
Inputs
  • a
  • b
  • c
Outputs
  • volume
  • surface
  • sphere
Formula
V = 4/3 πabc
Assumptions
  • Euclidean 3-space
  • Semi-axes a, b, c > 0
  • a=b=c is a valid sphere
  • Not a sphere page
  • Not the 2-D ellipse
  • Triaxial surface later
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • a ≤ 0 or b ≤ 0 or c ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
a=2 b=1.5 c=1 → V=4π
Validation cases

3 published on this page · 66/66 tests · Production surface contract 3/3 · View evidence

  • a=2 b=1.5 c=1 → V=4π sphere=false
  • a=1 b=1 c=1 → V=4π/3 A=4π sphere=true
  • a=2 b=1.5 c=0 → VALUE_MUST_BE_POSITIVE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.11.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dodecahedron Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dodecahedron Calculator Regular dodecahedron V = (1/4)(15+7√5)a³ and A = 3√(25+10√5) a² from one edge. Same engine as Solid Geometry. Not an icosahedron. Not an n-gon prism. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second solid engine?

No. It is mode=ellipsoid on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this the sphere calculator?

No. The sphere page stays on /calc/math/sphere. Equal semi-axes a=b=c are a valid ellipsoid that matches the sphere volume and surface.

Is this the 2-D ellipse?

No. The plane ellipse stays on /calc/math/ellipse. This page is a 3-D solid. Triaxial surface comes later.

Where does this run?

Locally in the browser by default. REST and MCP call the same solid-geometry engine.