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Math calculator

Ellipsoid Calculator

Ellipsoid V = 4/3 πabc and surface from the exact elliptic-integral model, numerically evaluated. 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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 21/21 declared partitions (sphere, cylinder, cone, torus, frustum, prism, pyramid, tetrahedron, octahedron, icosahedron, dodecahedron, regular-prism, regular-pyramid, regular-frustum, ellipsoid, spherical-cap, spherical-zone, spherical-wedge, spherical-lune, spherical-sector, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Triaxial ellipsoid volume and surface from the exact elliptic-integral model, numerically evaluated. Sphere surface when a=b=c.
Scope
Euclidean 3-space
Verification
Engine tested · Source checked · v1.18.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.18.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-21.solid-cvp-all-modes
Verification revision
2026-09-21.solid-cvp-all-modes · 4/4 property · digest d81459afdb09
Legacy regression
118/118 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
26/26 golden · 27/27 CVP boundary · 32/32 invalid · 4/4 property · 8/8 cross-interface · 2/2 cross-calculator · 3/3 CVP contract · Manifest
Sources
Sources
Implementation conventions
  • Modulus k, not m = k² — Incomplete F(φ,k) and E(φ,k) take elliptic modulus k. Passing m = k² as k is a known library footgun and is regression-tested.
  • Internal axis sort — Human inputs keep labels a, b, c. Surface uses max ≥ mid ≥ min. Permuting axes does not change V or A.
Methods
  • numeric_domain: real semi-axes a, b, c > 0
  • exactness: volume closed identity in float64; surface exact elliptic-integral / spheroid model, numerically evaluated
  • algorithm: sort p≥q≥r; sphere 4πp²; oblate artanh; prolate arcsin; else Carlson RF/RD → F(φ,k), E(φ,k) with modulus k
  • singularity_policy: a=b=c uses the sphere closed form so 1/sinφ is never evaluated at φ=0
  • independent_oracle: 2-D parametric surface Gauss–Legendre quadrature in tests; Legendre integrand F,E checks k vs m
  • golden_vectors: triaxial 2-1.5-1, sphere, oblate, prolate, six axis permutations, near-sphere, scale, high eccentricity
Evidence
43 legacy golden · 25 legacy boundary · legacy regression suite · 26/26 oracle-backed golden · 32/32 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.18.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

VolumeV = 4/3 πabc
Oblate spheroidA = 2πR²(1+((1−e²)/e) artanh e)
Prolate spheroidA = 2πc²(1+(R/(c e)) arcsin e)
Triaxial surfaceA = 2π(c² + ab/sinφ (E(φ,k)sin²φ + F(φ,k)cos²φ)) with modulus k, a≥b≥c after internal sort
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. Circular frustum stays /frustum. Incomplete elliptic F,E use modulus k (not parameter m=k²) and are internal to this mode, not a math.special page. Volume is the closed 4/3 πabc identity. Surface is the exact elliptic-integral / spheroid model, numerically evaluated. Not a second engine.

How to use

1

Enter semi-axes a, b, and c

All three semi-axes must be positive and use the same length unit. If a, b, c are in cm, V is in cm³ and A is in cm². Order does not matter; the engine sorts internally. Default a = 2, b = 1.5, c = 1.

2

Read volume and surface area

2-1.5-1 has V = 4π. Surface is the exact elliptic-integral model, numerically evaluated. 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

Oblate 2-2-1

a = 2, b = 2, c = 1

A = 2πR²(1+((1−e²)/e) artanh e)
≈34.6875308134
ϟ

Sphere identity

a = 1, b = 1, c = 1

V = 4π/3
4π/3

Ellipsoid calculator specification

Version 1.18.0 · 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. Surface uses spheroid closed forms when two axes are equal, otherwise the exact incomplete-elliptic expression in F(φ,k) and E(φ,k), numerically evaluated in float64. Inputs are sorted internally so a, b, c may be entered in any order. When a=b=c the solid is a sphere. This page is mode=ellipsoid on math.geometry.solid. Not /sphere. Not the 2-D ellipse. Not CAS.
What it calculates
Triaxial ellipsoid volume and surface from the exact elliptic-integral model, numerically evaluated. Sphere surface when a=b=c.
Inputs
  • a
  • b
  • c
Outputs
  • volume
  • surface
  • sphere
  • oblate
  • prolate
Formula
V = 4/3 πabc; spheroid closed forms or Legendre incomplete F(φ,k), E(φ,k) with modulus k
Assumptions
  • Euclidean 3-space
  • The Result Card includes an engine-linked 2-D generating section or silhouette of the published lengths. Not a 3-D mesh. Same pack as REST/MCP.
  • Semi-axes a, b, c > 0 in one consistent length unit; V has dimension length³ and A has dimension length²
  • Input order is a label only; surface uses internally sorted p ≥ q ≥ r
  • a=b=c is a valid sphere
  • Not a sphere page
  • Not the 2-D ellipse
  • Not a Knud Thomsen approximation
  • 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
Numerical precision
  • Volume is the closed identity 4/3 πabc evaluated in IEEE-754 binary64
  • Surface is float64 numerical evaluation of an exact model (spheroid closed forms or incomplete F,E)
  • Triaxial published surface agrees with independent 2-D parametric quadrature to 1e-9 relative at (2, 1.5, 1)
  • Classification uses exact axis equality; near-sphere stays on the triaxial path
Example
a=2 b=1.5 c=1 → V=4π A≈27.8864424735
Validation cases

6 published on this page · 118/118 tests · Production surface contract 5/5 · View evidence

  • a=2 b=1.5 c=1 → V=4π A≈27.8864424735 sphere=false
  • a=1 b=2 c=1.5 → same V and A as 2-1.5-1 (axis permutation)
  • a=2 b=2 c=1 → oblate closed form A≈34.6875308134
  • a=2 b=1 c=1 → prolate closed form
  • a=1 b=1 c=1 → V=4π/3 A=4π sphere=true
  • a=2 b=1.5 c=0 → VALUE_MUST_BE_POSITIVE
Methods
  • numeric_domain: real semi-axes a, b, c > 0
  • exactness: volume closed identity in float64; surface exact elliptic-integral / spheroid model, numerically evaluated
  • algorithm: sort p≥q≥r; sphere 4πp²; oblate artanh; prolate arcsin; else Carlson RF/RD → F(φ,k), E(φ,k) with modulus k
  • singularity_policy: a=b=c uses the sphere closed form so 1/sinφ is never evaluated at φ=0
  • independent_oracle: 2-D parametric surface Gauss–Legendre quadrature in tests; Legendre integrand F,E checks k vs m
  • golden_vectors: triaxial 2-1.5-1, sphere, oblate, prolate, six axis permutations, near-sphere, scale, high eccentricity
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Quantity names, symbols, and the volume identity V = 4/3 πabc
  • NIST DLMF §19.2 Incomplete elliptic integrals — 19.2(ii) Legendre’s integrals F(φ,k) and E(φ,k) · accessed 2026-09-21
    Supports: Production incomplete F and E use modulus k, not parameter m = k²
  • John D. Cook — Ellipsoid surface area · 2014-07-06 · accessed 2026-09-21
    Supports: Triaxial surface in incomplete F(φ,k), E(φ,k) with a ≥ b ≥ c after sorting; warns that many libraries take m = k²
Implementation conventions
  • Modulus k, not m = k² — Incomplete F(φ,k) and E(φ,k) take elliptic modulus k. Passing m = k² as k is a known library footgun and is regression-tested.
  • Internal axis sort — Human inputs keep labels a, b, c. Surface uses max ≥ mid ≥ min. Permuting axes does not change V or A.
Calculation version
1.18.0

Other calculators in this family: Cone Calculator, Cylinder Calculator, Dodecahedron Calculator, Frustum Calculator, Icosahedron Calculator, Octahedron Calculator, Prism Calculator, Pyramid Calculator . Explore all Solid Geometry.

→ 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. › → 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. › → Frustum Calculator Right circular frustum V = (1/3)πh(R²+Rr+r²) from two radii and height. Same engine as Solid Geometry. Not /cone. Not ISO 5167-5. Not CAS. Runs locally. › → Icosahedron Calculator Regular icosahedron V = (5/12)(3+√5)a³ and A = 5√3 a² from one edge. Same engine as Solid Geometry. Not an octahedron. Not a dodecahedron. Not CAS. Runs locally. › → Octahedron Calculator Regular octahedron V = (√2/3)a³ and A = 2√3 a² from one edge. Same engine as Solid Geometry. Not a tetrahedron. Not an icosahedron. Not CAS. Runs locally. › → Prism Calculator Right rectangular prism V = abh from three edge lengths. Same engine as Solid Geometry. Not a frustum. Not a pyramid. Not CAS. Runs locally. › → Pyramid Calculator Right rectangular pyramid V = abh/3 from three lengths. Same engine as Solid Geometry. Not a prism. Not a tetrahedron. Not CAS. Runs locally. ›

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. Surface uses elliptic integrals internally; that is not a math.special elliptic page.

Where does this run?

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