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Math calculator

Arc Calculator

Circle arc s = r θ from radius and central angle. Same engine as Circle. Not /sector. Not /spherical-sector. Not CAS. 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 · 7/7 declared partitions (from-radius, from-diameter, from-circumference, from-area, arc, sector, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Circular arc length from radius and central angle, with an engine-linked arc.
Scope
Euclidean plane
Verification
Engine tested · Specification checked · v1.1.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.geometry-cvp
Verification revision
2026-09-22.geometry-cvp · 4/4 property · digest 9db7bf18f0a0
Legacy regression
10/10 tests · Production surface contract 5/5
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
6/6 golden · 3/3 CVP boundary · 7/7 invalid · 4/4 property · 4/4 cross-interface · 3/3 cross-calculator · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 6/6 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.0 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.

Arc lengths = r θ
SectorA = (1/2) r² θ
iSame engine as /circle. Default r=2, θ=180° gives s=2π. θ is entered in degrees. Not a second engine.

How to use

1

Enter r and θ

Radius must be positive. Central angle is in degrees. Default r=2, θ=180°.

2

Read arc length

A semicircle of radius 2 is 2π. The arc updates with r and θ.

Example calculations

Common configurations with formula and result.

ϟ

Semicircle

r = 2, θ = 180°

s = 2π
2π
ϟ

Quarter circle

r = 2, θ = 90°

s = π
π

Arc calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A circular arc of radius r>0 and central angle θ has length rθ with θ in radians. This page is mode=arc on math.geometry.circle. Not /sector. Not /spherical-sector. Not CAS.
What it calculates
Circular arc length from radius and central angle, with an engine-linked arc.
Inputs
  • r
  • theta_deg
Outputs
  • arc_length
  • sector_area
  • r
  • theta_deg
  • theta_rad
Formula
s = r θ
Assumptions
  • Euclidean plane
  • θ entered in degrees, used in radians
  • Not /sector
  • Not /spherical-sector
  • Not CAS
Units
  • length (consistent units)
  • degrees
Boundary conditions
  • r ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
r=2 theta_deg=180 → s=2π
Validation cases

2 published on this page · 10/10 tests · Production surface contract 5/5 · View evidence

  • r=2 theta_deg=180 → s=2π
  • r=2 theta_deg=90 → s=π
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Annulus Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola Calculator, Parallelogram Calculator . Explore all Geometry.

→ Annulus Calculator Annulus A = π(R² − r²) and P = 2π(R + r) with outer R > inner r. Same engine as Plane Geometry. Not /circle. Not a square. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Discovery /sector, /arc. Not the Pythagorean page. Not /spherical-sector. 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. › → Ellipse Calculator Standard ellipse x²/a² + y²/b² = 1. Same engine as Conic Section. a = b is a valid ellipse with e = 0; the circle-only page is /circle. Not CAS. Runs locally. › → 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. › → Irregular Polygon Calculator Shoelace area of a simple 3–8 vertex polygon. Same engine as Plane Geometry. Not /regular-polygon. Not a triangle solver. Not CAS. Runs locally. › → Parabola Calculator Standard parabola x² = 4py (vertical) or y² = 4px (horizontal). Same engine as Conic Section. p ≠ 0. Not CAS. Runs locally. › → Parallelogram Calculator Parallelogram A = ab sin θ, P = 2(a+b), diagonals from a, b, and included angle. Same engine as Plane Geometry. Not a rectangle page. Not the fluid trapezoid. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second circle engine?

No. It is mode=arc on math.geometry.circle, the same engine as /calc/math/circle.

Is this the sector page?

No. Sector area stays on /calc/math/sector. Both share this engine.

Where does this run?

Locally in the browser by default. REST and MCP call the same circle engine.