HomeCalculatorsMathGeometryConic Section Calculator
Math calculator

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.

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
Axis-aligned ellipse, parabola, and hyperbola in standard form.
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

Ellipsex²/a² + y²/b² = 1
Parabolax² = 4py
Hyperbolax²/a² − y²/b² = 1
iEllipse perimeter is Ramanujan I, not a complete elliptic integral. Discovery pages lock one mode. Not coordinate geometry. Not CAS.

How to use

1

Choose ellipse, parabola, or hyperbola

Ellipse and hyperbola need a > 0 and b > 0. Parabola needs p ≠ 0 and vertical or horizontal orientation.

2

Read foci, eccentricity, and the equation

a = 5, b = 4 is the 5-4-3 ellipse. p = 1 vertical focuses at (0, 1). a = 3, b = 4 hyperbola has c = 5.

Example calculations

Common configurations with formula and result.

ϟ

5-4 ellipse

a = 5, b = 4

c = √(a² − b²)
c=3, e=0.6
ϟ

Vertical parabola

p = 1

x² = 4py
focus (0, 1)
ϟ

3-4-5 hyperbola

a = 3, b = 4

c = √(a² + b²)
c=5

Conic Section calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Axis-aligned standard forms: ellipse x²/a² + y²/b² = 1, parabola x² = 4py or y² = 4px, hyperbola x²/a² − y²/b² = 1 (or the vertical swap). a = b is a valid ellipse with e = 0; the circle-only page stays /calc/math/circle. Not rotated (no xy term). Not CAS.
What it calculates
Axis-aligned ellipse, parabola, and hyperbola in standard form.
Inputs
  • mode
  • a?
  • b?
  • p?
  • orientation?
Outputs
  • equation
  • c
  • e
  • area
  • focus_x
  • focus_y
Formula
x²/a² + y²/b² = 1; x² = 4py; x²/a² − y²/b² = 1
Assumptions
  • Euclidean plane
  • Axis-aligned standard forms
  • Not rotated (no xy term)
  • Not translated
  • Not the circle-only page
  • Not coordinate geometry
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE
  • parabola p = 0 → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
ellipse a=5 b=4 → c=3, e=0.6
Validation cases

3 published on this page · 10/10 tests · Production surface contract 3/3 · View evidence

  • mode=ellipse a=5 b=4 → c=3 e=0.6
  • mode=parabola p=1 → focus (0, 1)
  • mode=hyperbola a=3 b=4 → c=5
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, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product Calculator, Ellipse 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. › → 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. › → 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. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is a = b the circle calculator?

No. a = b is a valid ellipse with e = 0. The circle-only page is /calc/math/circle.

Is this coordinate geometry?

No. Distance, midpoint, slope, two-point line, intersection, and angle stay on /calc/math/coordinate-geometry.

Where does this run?

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