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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 Engine tested · Specification checked · 10/10 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Standard ellipse foci, eccentricity, area, and Ramanujan I perimeter.
Scope
Euclidean plane
Verification
Engine tested · 10/10 tests · Production surface contract 3/3 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

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.

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, and Ramanujan I perimeter.
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 3/3 · 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: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product 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. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. ›
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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.