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Math calculator

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.

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 hyperbola foci, eccentricity, and asymptote slope.
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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
Focic = √(a² + b²)
Eccentricitye = c/a
iDefault orientation is horizontal. Same engine as /conic-section. Not the ellipse page. Not CAS.

How to use

1

Enter a and b

Both must be positive. a = 3, b = 4 is the 3-4-5 hyperbola with e = 5/3.

2

Read c, e, and the asymptote slope

Horizontal foci are (±c, 0). Vertical foci are (0, ±c).

Example calculations

Common configurations with formula and result.

ϟ

3-4-5 hyperbola

a = 3, b = 4, horizontal

c = √(9 + 16)
c=5, e=5/3
ϟ

Vertical

a = 3, b = 4, vertical

y²/a² − x²/b² = 1
c=5

Hyperbola calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Horizontal hyperbola x²/a² − y²/b² = 1 with a > 0, b > 0, c = √(a² + b²), e = c/a > 1, asymptotes y = ±(b/a)x. Vertical is y²/a² − x²/b² = 1. This page is mode=hyperbola on math.geometry.conic. Not a second engine. Not CAS.
What it calculates
Standard hyperbola foci, eccentricity, and asymptote slope.
Inputs
  • a
  • b
  • orientation?
Outputs
  • c
  • e
  • asymptote_slope
  • equation
Formula
x²/a² − y²/b² = 1
Assumptions
  • Euclidean plane
  • Centered at the origin
  • Axis-aligned
  • Not ellipse
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
a=3 b=4 → c=5, e=5/3
Validation cases

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

  • a=3 b=4 → c=5
  • a=3 b=4 orientation=vertical → 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, 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=hyperbola on math.geometry.conic, the same engine as /calc/math/conic-section.

Is this the ellipse calculator?

No. Ellipse eccentricity is in [0, 1). Hyperbola eccentricity is greater than 1. Ellipse stays on /calc/math/ellipse.

Where does this run?

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