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Math calculator

Parabola Calculator

Standard parabola x² = 4py (vertical) or y² = 4px (horizontal). Same engine as Conic Section. p ≠ 0. 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 parabola focus, directrix, equation, and an engine-linked figure from focal length p.
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
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.

Verticalx² = 4py
FocusF = (0, p)
Directrixy = −p
iHorizontal is y² = 4px. Same engine as /conic-section. Not a quadratic-equation page. Not CAS.

How to use

1

Enter p and orientation

p = 1 vertical focuses at (0, 1) with directrix y = −1. p = 0 is invalid.

2

Read focus, directrix, and the equation

The vertex is the origin. The figure shows the focus, directrix, and latus rectum. The quadratic formula page stays at /calc/math/quadratic.

Example calculations

Common configurations with formula and result.

ϟ

Vertical p = 1

orientation vertical

x² = 4y
focus (0, 1)
ϟ

Horizontal p = 2

orientation horizontal

y² = 8x
focus (2, 0)

Parabola calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A parabola with focal length p is x² = 4py (vertical, focus (0, p), directrix y = −p) or y² = 4px (horizontal, focus (p, 0), directrix x = −p). This page is mode=parabola on math.geometry.conic. p ≠ 0. Not a second engine. Not CAS.
What it calculates
Standard parabola focus, directrix, equation, and an engine-linked figure from focal length p.
Inputs
  • p
  • orientation?
Outputs
  • focus_x
  • focus_y
  • directrix
  • equation
Formula
x² = 4py
Assumptions
  • Euclidean plane
  • Vertex at the origin
  • Axis-aligned
  • Not quadratic roots
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • p = 0 → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
p=1 vertical → focus (0, 1)
Validation cases

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

  • p=1 orientation=vertical → focus (0, 1)
  • p=2 orientation=horizontal → focus (2, 0)
Sources
Calculation version
1.0.0

Background

Interpretation and common distinctions.

What does this parabola calculator compute?

A parabola with focal length p is x² = 4py (vertical, focus (0, p), directrix y = −p) or y² = 4px (horizontal, focus (p, 0), directrix x = −p).

The published formula or method is x² = 4py. The following references support the stated mathematics or method:

Which inputs and limits should you check?

Use the indicated standard-form orientation and positive geometric parameters. A numerical circumference approximation and an exact defining equation describe different aspects of a conic.

The worked example uses orientation vertical with x² = 4y. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.conic v1.0.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon 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 Calculate circular arc length from radius and central angle in degrees. See the radian formula, worked examples, units, and mathematical sources. › → Circle Calculator Calculate circle radius, diameter, circumference, area, arc length, and sector area. Explore formulas, worked examples, units, and mathematical sources. › → 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. › → 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. › → 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=parabola on math.geometry.conic, the same engine as /calc/math/conic-section.

Is this the quadratic formula?

No. Roots of ax² + bx + c = 0 stay on /calc/math/quadratic. This page is the geometric parabola.

Where does this run?

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