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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 Engine tested · Specification checked · 10/10 tests · Production surface contract 3/3 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
Circular arc length from radius and central angle.
Scope
Euclidean plane
Verification
Engine tested · 10/10 tests · Production surface contract 3/3 · Specification checked · v1.1.0
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: STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (3 capabilities; 161 remain CURRENT) @ 2026-09-20T23:37:12.315Z
Semantic contract
PASS

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.

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.
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 3/3 · 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: Angle Between Lines Calculator, Annulus Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder 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. › → 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. › → 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. ›
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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.