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Math calculator

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.

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
The smaller (principal) angle between two plane lines, each given by two distinct points, with an engine-linked figure of that angle.
Scope
Euclidean plane
Verification
Engine tested · Specification 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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
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.

Principal angleθ = arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²)))
Range0° ≤ θ ≤ 90°
Line from two pointsA = y₂ − y₁, B = x₁ − x₂, C = x₂ y₁ − y₂ x₁
iSame engine as /coordinate-geometry. Intersection stays on /line-intersection. Two-point line stays on /line-equation.

How to use

1

Enter the first line as two distinct points

(0, 0) and (2, 2) is y = x.

2

Enter the second line as two distinct points

(0, 0) and (2, 0) is y = 0. The principal angle is 45°.

Example calculations

Common configurations with formula and result.

ϟ

45°

(0, 0)–(2, 2) and (0, 0)–(2, 0)

arccos(1/√2)
45°
ϟ

Perpendicular

(0, 0)–(2, 0) and (0, 0)–(0, 2)

n₁ · n₂ = 0
90°
ϟ

Parallel

(0, 0)–(2, 0) and (0, 1)–(2, 1)

Δ = 0
0°

Angle Between Lines calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The principal angle θ between two undirected lines A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0 is arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²))), in [0°, 90°]. For oblique lines this is the acute angle. This page is mode=angle on math.geometry.coordinate. Each line is two distinct points. Parallel and coincident lines are 0°. Perpendicular lines are 90°.
What it calculates
The smaller (principal) angle between two plane lines, each given by two distinct points, with an engine-linked figure of that angle.
Inputs
  • x1
  • y1
  • x2
  • y2
  • x3
  • y3
  • x4
  • y4
Outputs
  • angle_deg
  • angle_rad
  • status
  • supplement_deg
  • standard1
  • standard2
Formula
θ = arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²)))
Assumptions
  • Euclidean plane
  • Two distinct points per line
  • Undirected lines (principal angle in [0°, 90°])
  • Not 3-D / spherical
  • Not Pythagorean
  • Not CAS
Units
  • degree
  • radian
Boundary conditions
  • missing coordinates → MISSING_REQUIRED_INPUT
  • coincident points on a line → INVALID_INPUT
  • parallel distinct → status=parallel, 0°
  • same line → status=coincident, 0°
  • unknown mode (including cas, pythagorean) → INVALID_MODE
Example
x1=0 y1=0 x2=2 y2=2 x3=0 y3=0 x4=2 y4=0 → 45°
Validation cases

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

  • x1=0 y1=0 x2=2 y2=2 x3=0 y3=0 x4=2 y4=0 → 45°
  • mode=cas x1=0 y1=0 x2=2 y2=2 x3=0 y3=0 x4=2 y4=0 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.5.0

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

→ 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. › → 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 a second coordinate engine?

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

Is the result the acute angle?

For oblique lines, yes. The principal line angle is always in [0°, 90°]. 0° is parallel or coincident, not acute. 90° is a right angle, not acute. The supplement is 180° − θ.

What if a pair of points coincides?

That pair does not determine a unique line. The engine returns INVALID_INPUT.