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Math calculator

Ellipse Calculator

Standard ellipse x²/a² + y²/b² = 1. Same engine as Conic Section. a = b is a valid ellipse with e = 0; the circle-only page is /circle. 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 · 4/4 declared partitions (ellipse, parabola, hyperbola, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Standard ellipse foci, eccentricity, area, Ramanujan I perimeter, and an engine-linked figure of the axes and foci.
Scope
Euclidean plane
Verification
Engine tested · Specification checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.geometry-cvp
Verification revision
2026-09-22.geometry-cvp · 3/3 property · digest 7a8a807960ea
Legacy regression
10/10 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 1/1 (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
6/6 golden · 2/2 CVP boundary · 6/6 invalid · 3/3 property · 4/4 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 6/6 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 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.

Standardx²/a² + y²/b² = 1
Eccentricitye = c / max(a, b)
AreaA = πab
iPerimeter is Ramanujan I: π[3(a+b) − √((3a+b)(a+3b))], not a complete elliptic integral. Same engine as /conic-section. Not CAS.

How to use

1

Enter a and b

Both must be positive. a = 5, b = 4 is the 5-4-3 ellipse. a = b is a circle on this engine, not the circle-only page.

2

Read c, e, area, and perimeter

Foci sit at (±c, 0) when a > b, or (0, ±c) when b > a. The figure updates with a and b.

Example calculations

Common configurations with formula and result.

ϟ

5-4 ellipse

a = 5, b = 4

c = √(25 − 16)
c=3, e=0.6
ϟ

Unit circle as ellipse

a = 1, b = 1

e = 0
e=0, A=π

Ellipse calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
x²/a² + y²/b² = 1 with a > 0, b > 0. c = √|a² − b²|, e = c / max(a, b), area = πab. Perimeter uses Ramanujan I. This page is mode=ellipse on math.geometry.conic. a = b is a circle with e = 0; /calc/math/circle stays the circle-only page. Not a second engine. Not CAS.
What it calculates
Standard ellipse foci, eccentricity, area, Ramanujan I perimeter, and an engine-linked figure of the axes and foci.
Inputs
  • a
  • b
Outputs
  • c
  • e
  • area
  • perimeter
  • equation
  • circle
Formula
x²/a² + y²/b² = 1
Assumptions
  • Euclidean plane
  • Centered at the origin
  • Axes aligned with x and y
  • Not the circle-only page
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
a=5 b=4 → c=3, e=0.6
Validation cases

2 published on this page · 10/10 tests · Production surface contract 5/5 · View evidence

  • a=5 b=4 → c=3 e=0.6
  • a=1 b=1 → e=0 circle=true
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Conic Section Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola Calculator, Parallelogram Calculator . Explore all Geometry.

→ Annulus Calculator Annulus A = π(R² − r²) and P = 2π(R + r) with outer R > inner r. Same engine as Plane Geometry. Not /circle. Not a square. Not CAS. Runs locally. › → Arc Calculator Circle arc s = r θ from radius and central angle. Same engine as Circle. Not /sector. Not /spherical-sector. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Discovery /sector, /arc. Not the Pythagorean page. Not /spherical-sector. 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. › → Hyperbola Calculator Standard hyperbola x²/a² − y²/b² = 1 (horizontal) or y²/a² − x²/b² = 1 (vertical). Same engine as Conic Section. Not CAS. Runs locally. › → Irregular Polygon Calculator Shoelace area of a simple 3–8 vertex polygon. Same engine as Plane Geometry. Not /regular-polygon. Not a triangle solver. Not CAS. Runs locally. › → Parabola Calculator Standard parabola x² = 4py (vertical) or y² = 4px (horizontal). Same engine as Conic Section. p ≠ 0. Not CAS. Runs locally. › → Parallelogram Calculator Parallelogram A = ab sin θ, P = 2(a+b), diagonals from a, b, and included angle. Same engine as Plane Geometry. Not a rectangle page. Not the fluid trapezoid. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second conic engine?

No. It is mode=ellipse on math.geometry.conic, the same engine as /calc/math/conic-section.

Is a = b the circle calculator?

No. a = b is a valid ellipse with e = 0. Diameter, circumference, arc, and sector stay on /calc/math/circle.

Where does this run?

Locally in the browser by default. REST and MCP call the same conic-section engine.