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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 Engine tested · Specification checked · 10/10 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Standard parabola focus, directrix, and equation from focal length p.
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

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 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, and equation 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 3/3 · View evidence

  • p=1 orientation=vertical → focus (0, 1)
  • p=2 orientation=horizontal → focus (2, 0)
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=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.