HomeCalculatorsMathGeometryPoint Line Distance Calculator
Math calculator

Point Line Distance Calculator

Perpendicular distance from a plane point to the line through two points. Same engine as Coordinate Geometry. Coincident line points are invalid. Not Pythagorean. Not a second engine. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 11/11 tests · Production surface contract 3/3 · v1.4.0
Input interpretation
Enter values to calculate.
Result
—
Model
Perpendicular distance from a plane point to the unique line through two distinct points.
Scope
Euclidean plane
Verification
Engine tested · 11/11 tests · Production surface contract 3/3 · Specification checked · v1.4.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
5 golden · 1 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.4.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · 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.

Point-lined = |Ax₀ + By₀ + C| / √(A² + B²)
Line from two pointsA = y₂ − y₁, B = x₁ − x₂, C = x₂ y₁ − y₂ x₁
Foot(x₀, y₀) − t(A, B)
iSame engine as /coordinate-geometry. Two-point line stays on /line-equation. Vertical lines work. Not CAS.

How to use

1

Enter the line as two distinct points

(0, 0) and (4, 0) is the x-axis. The two-point equation page stays at /line-equation.

2

Enter the query point

(1, 3) is 3 units from the x-axis. The foot is (1, 0).

Example calculations

Common configurations with formula and result.

ϟ

Distance 3

Line (0, 0)–(4, 0), point (1, 3)

|3| / 1
3
ϟ

√2 to x + y = 2

Line (2, 0)–(0, 2), point (0, 0)

2 / √2
1.41421356237
ϟ

On the line

Line (0, 0)–(4, 0), point (2, 0)

|0| / 1
0

Point Line Distance calculator specification

Version 1.4.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The distance from (x₀, y₀) to Ax + By + C = 0 is |Ax₀ + By₀ + C| / √(A² + B²). This page is mode=point_line on math.geometry.coordinate. The line is the unique line through two distinct points. Coincident points are invalid. Not a second engine. Not Pythagorean. Not CAS.
What it calculates
Perpendicular distance from a plane point to the unique line through two distinct points.
Inputs
  • x1
  • y1
  • x2
  • y2
  • x0
  • y0
Outputs
  • distance
  • signed
  • foot_x
  • foot_y
  • A
  • B
  • C
  • standard
Formula
|Ax₀ + By₀ + C| / √(A² + B²)
Assumptions
  • Euclidean plane
  • Two distinct line points
  • Not 3-D / spherical
  • Not Pythagorean
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • missing coordinates → MISSING_REQUIRED_INPUT
  • coincident line points → INVALID_INPUT
  • unknown mode (including cas, pythagorean) → INVALID_MODE
Example
x1=0 y1=0 x2=4 y2=0 x0=1 y0=3 → 3
Validation cases

2 published on this page · 11/11 tests · Production surface contract 3/3 · View evidence

  • x1=0 y1=0 x2=4 y2=0 x0=1 y0=3 → distance=3
  • mode=cas x1=0 y1=0 x2=4 y2=0 x0=1 y0=3 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.4.0

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 coordinate engine?

No. It is mode=point_line on math.geometry.coordinate, the same engine as /calc/math/coordinate-geometry.

What if the two line points coincide?

There is no unique line. The engine returns INVALID_INPUT.

Where is the two-point line equation?

/calc/math/line-equation is mode=line on this same engine.