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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 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
Axis-aligned ellipse, parabola, and hyperbola in standard form, with an engine-linked figure of foci, directrix, or asymptotes.
Scope
Euclidean plane
Verification
Engine tested · Source 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
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • S. Ramanujan — Modular Equations and Approximations to π (1914)
Sources
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.

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. The figure tracks the same engine result.

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, with an engine-linked figure of foci, directrix, or asymptotes.
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 5/5 · 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
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Quantity names and symbols for ellipse, parabola, and hyperbola
  • S. Ramanujan — Modular Equations and Approximations to π (1914)
    Supports: Ellipse perimeter approximation I: π[3(a+b)−√((3a+b)(a+3b))]; not an elliptic integral
Calculation version
1.0.0

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Ellipse 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. › → 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. › → 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 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.