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Math calculator

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.

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 · 8/8 declared partitions (distance, midpoint, slope, line, point-line, intersect, angle, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Distance, midpoint, and slope of two plane points, with an engine-linked figure of the segment.
Scope
Euclidean plane
Verification
Engine tested · Source checked · v1.5.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.5.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.geometry-cvp
Verification revision
2026-09-22.geometry-cvp · 3/3 property · digest c0f46759e9a3
Legacy regression
21/21 tests · Production surface contract 6/6
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
10/10 golden · 4/4 CVP boundary · 8/8 invalid · 3/3 property · 4/4 cross-interface · 2/2 cross-calculator · 6/6 CVP contract · Manifest
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — analytic geometry
Sources
Evidence
8 legacy golden · 3 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 8/8 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.5.0 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.

Coordinate geometry primitivesDistance, midpoint, slope, two-point line, point-line distance, line intersection, and principal angle between lines
Distanced = √((x₂−x₁)²+(y₂−y₁)²)
MidpointM = ((x₁+x₂)/2, (y₁+y₂)/2)
Slopeslope = (y₂−y₁)/(x₂−x₁)
Two-point lineA = y₂ − y₁, B = x₁ − x₂, C = x₂ y₁ − y₂ x₁
Point-line distance|Ax₀ + By₀ + C| / √(A² + B²)
Line intersectionA₁x + B₁y + C₁ = 0 ∩ A₂x + B₂y + C₂ = 0
Angle between linesθ = arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²))), 0° ≤ θ ≤ 90°
iShare URLs use x1,y1,x2,y2 — not m (reserved). Vertical lines report undefined slope. The line through the same two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines.

How to use

1

Choose distance, midpoint, or slope

Enter both points. Slope of a vertical line is undefined.

2

Read the scalar or the point

Distance of (0,0) to (3,4) is 5, the same number as a 3-4-5 triangle, on a different model.

Example calculations

Common configurations with formula and result.

ϟ

3-4-5 distance

(0,0) to (3,4)

√(9+16)
5
ϟ

Slope 2

(0,1) to (2,5)

4/2
2

Coordinate Geometry calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The Euclidean distance between (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)²+(y₂−y₁)²). The midpoint is the average of the coordinates. Slope is Δy/Δx when the line is not vertical.
What it calculates
Distance, midpoint, and slope of two plane points, with an engine-linked figure of the segment.
Inputs
  • mode
  • x1
  • y1
  • x2
  • y2
Outputs
  • distance
  • mx
  • my
  • slope
  • vertical
Formula
d = √((x₂−x₁)²+(y₂−y₁)²)
Assumptions
  • Euclidean plane
  • Not 3-D / spherical
Units
  • length (consistent units)
Boundary conditions
  • missing coordinates → MISSING_REQUIRED_INPUT
  • unknown mode → INVALID_MODE
Example
distance (0,0)(3,4) → 5
Validation cases

5 published on this page · 21/21 tests · Production surface contract 6/6 · View evidence

  • mode=distance x1=0 y1=0 x2=3 y2=4 → distance=5
  • mode=slope x1=2 y1=0 x2=2 y2=5 → vertical=true
  • mode=point_line x1=0 y1=0 x2=4 y2=0 x0=1 y0=3 → distance=3
  • mode=intersect x1=0 y1=0 x2=2 y2=2 x3=0 y3=2 x4=2 y4=0 → (1, 1)
  • mode=angle x1=0 y1=0 x2=2 y2=2 x3=0 y3=0 x4=2 y4=0 → 45°
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Quantity names and symbols for plane coordinates, distance, and angle
  • CRC Standard Mathematical Tables — analytic geometry
    Supports: Distance √(Δx²+Δy²); point–line |Ax₀+By₀+C|/√(A²+B²); undirected line angle via atan2(|d1×d2|, |d1·d2|)
Calculation version
1.5.0

Other calculators in this family: Angle Between Lines Calculator, Line Equation Calculator, Line Intersection Calculator, Point Line Distance Calculator . Explore all Analytic Geometry.

→ Angle Between Lines Calculator Find the smaller angle between two 2D lines, each defined by two points. Parallel or coincident lines give 0°; perpendicular lines give 90°. Runs locally. › → Line Equation Calculator Equation of the line through two plane points. Same engine as Coordinate Geometry. Vertical is x = x₁. Coincident points are invalid. Not Pythagorean. Not a second engine. Not CAS. Runs locally. › → Line Intersection Calculator Intersection of two plane lines, each given by two points. Parallel distinct lines are empty. Coincident lines are infinite. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → 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. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this the Pythagorean calculator?

No. This page takes coordinates. /calc/math/pythagorean solves a right triangle from two sides.

Where is the line through two points?

/calc/math/line-equation is mode=line on this same engine. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines.

Where does this run?

Locally in the browser by default. REST and MCP call the same coordinate-geometry engine.