Arc Calculator
Calculate circular arc length from radius and central angle in degrees. See the radian formula, worked examples, units, and mathematical sources.
Enter values to calculate.
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 · Source checked · v1.2.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
- Versions
- Calculation 1.2.0 · CVP protocol 1.0.0-proposed · Evidence 2026-10-29.math-uuce-fields-v1
- Verification revision
- 2026-10-29.math-uuce-fields-v1 · 4/4 property · digest 518c456dfeec
- 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
- Supplemental domain review
- Not performed
- Named expert review
- Not performed
- CVP suite
- 10/10 golden · 6/6 CVP boundary · 14/14 invalid · 4/4 property · 4/4 metamorphic · 15/15 cross-interface · 3/3 cross-calculator · 5/5 CVP contract · Manifest
- Sources
- Gilbert Strang, Calculus — §§1.4–1.5 Circular Motion and A Review of Trigonometry
- Gilbert Strang, Calculus — §§9.1–9.3 Polar Coordinates, Length and Area
- ISO 80000-2:2019 Quantities and units — Mathematics
- Evidence
- 3 legacy golden · 2 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 14/14 invalid · Artifact integrity PASS
- Semantic contract
- PASS
Full verification
Formulas
Core equations used by this calculator.
How to use
Enter r and θ
Enter a positive radius and the central angle in degrees. For an ordinary arc on one circle, use an angle from 0° to 360°.
Read arc length
Read the curved distance in the same unit as the radius. A radius of 2 cm and angle of 180° give 2π cm, approximately 6.28318530718 cm.
Example calculations
Common configurations with formula and result.
Semicircle
r = 2, θ = 180°
Quarter circle
r = 2, θ = 90°
Arc calculator specification
Version 1.2.0 · Engine tested
- Engine tested 10/10 tests · Production surface contract 5/5
- Named expert review Not performed
- Calculation version 1.2.0
- Definition
- Arc length is the distance along a circle's curved boundary. For radius r and central angle θ in radians, s = rθ. Enter the angle in degrees; the calculator converts it to radians.
- 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
- Radius and arc length use the same length unit; sector area uses its square
- Signed angles and multiple turns are preserved in the calculation; the diagram shows at most one turn
- Units
- length (consistent units)
- degrees
- Boundary conditions
- Raw inputs accept JSON numbers or decimal strings. Missing active fields return MISSING_REQUIRED_INPUT; nonzero decimal text converted to zero returns NUMERIC_UNDERFLOW.
- Unknown fields return UNKNOWN_INPUT. Radius/r, diameter/d and theta/theta_deg aliases must agree or return AMBIGUOUS_INPUT. Normalized responses contain only the selected mode and its active numeric fields.
- Nonfinite, underflowed or subnormal outputs → NUMERIC_RANGE
- r ≤ 0 → VALUE_MUST_BE_POSITIVE
- unknown mode → INVALID_MODE
- Numerical precision
- Small relative radius errors affect circumference linearly and disk area at twice that relative scale; sector area also depends linearly on its nonzero angle. Near zero angle use absolute error, not relative error.
- Version pins reject unsupported remote versions. Shared pages use the current engine with a version notice; historical execution and cross-runtime bitwise replay are not promised.
- 12 significant digits; supported nonzero magnitudes are preserved without an absolute zero threshold
- All nonzero outputs must be finite normal binary64 values (magnitude at least 2^-1022); exact zero angles are supported. Outside this envelope, NUMERIC_RANGE requests rescaling instead of returning zero, Infinity, or NaN.
- 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
- Sources
- Gilbert Strang, Calculus — §§1.4–1.5 Circular Motion and A Review of Trigonometry — §§1.4–1.5 Circular Motion and A Review of TrigonometrySupports: Radians, the equivalence of 360° and 2π radians, and circular circumference 2πr. Scaling the unit-circle distance gives arc length s = rθ; right-triangle sine gives chord length 2r sin(θ/2). Signed angles and multi-turn behavior are CalculatorX’s explicit policy. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
- Gilbert Strang, Calculus — §§9.1–9.3 Polar Coordinates, Length and Area — §§9.1–9.3 Polar Coordinates, Length and AreaSupports: Polar coordinates and the circular wedge area r²Δθ/2, together with area = ∫r² dθ/2 and polar arc-length integration. A full turn gives disk area πr² and circumference 2πr. An annulus follows by subtracting disk areas and adding both boundary lengths; these are derived applications. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
- ISO 80000-2:2019 Quantities and units — MathematicsSupports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
- Gilbert Strang, Calculus — §§1.4–1.5 Circular Motion and A Review of Trigonometry — §§1.4–1.5 Circular Motion and A Review of Trigonometry
- Calculation version
- 1.2.0
Background
Interpretation and common distinctions.
What is arc length?
Arc length is the distance along the curved edge of a circle between two endpoints. For radius r and central angle θ in radians, its length is s = rθ. If the angle is in degrees, use s = r × θ_deg × π/180.
Radians and circular motion are explained in Gilbert Strang’s Calculus, §§1.4–1.5, hosted by MIT OpenCourseWare. Scaling the unit-circle distance by radius r gives s = rθ. The formula applies to a circular arc in a flat plane; other curves require their own arc-length model.
How do you calculate an arc from a degree angle?
Convert the central angle to radians, then multiply by the radius. For a semicircle with r = 2 cm and θ = 180°, θ_rad = π and s = 2π cm ≈ 6.28318530718 cm.
| Central angle | Radians | Arc length when r = 2 cm |
|---|---|---|
| 0° | 0 | 0 cm |
| 90° | π/2 | π cm |
| 180° | π | 2π cm |
| 360° | 2π | 4π cm |
Arc length uses the radius's length unit. Entering a radius in meters gives a result in meters; entering it in inches gives inches. Displayed decimals are numerical approximations, while the π-based expressions above are exact mathematical values.
How does arc length differ from chord length and sector area?
Arc length measures a curved distance; chord length measures the straight line joining the endpoints; sector area measures the enclosed slice of the disk. These are different quantities, even when they use the same radius and angle.
For an arc from 0° to 360°, with θ expressed in radians, s = rθ, chord = 2r sin(θ/2), and sector area = r²θ/2. Strang’s polar-coordinate area chapter, §9.3, hosted by MIT OpenCourseWare gives the circular wedge formula r²θ/2; the arc formula follows from radian measure. The chord formula follows by bisecting the isosceles triangle formed by two radii and applying the sine relation in Strang’s trigonometry review, §1.5; for angles above 180°, use the complementary minor angle to construct that triangle. This page reports arc length and the corresponding sector area; the Sector calculator emphasizes the area.
How are negative angles and extra turns interpreted?
A nonnegative angle from 0° to 360° describes an ordinary circular arc. This calculator also preserves signed angles and multiple turns: a negative angle gives a directed, signed result, and 720° gives a path length of two circumferences.
The diagram displays at most one complete turn. For a physical distance along a single circle, enter the desired nonnegative sweep; do not interpret a negative numerical result as a negative physical distance.
What supports the formula and the result?
The linked MIT-hosted textbook chapters support radian measure, the sector-area formula and the chord derivation. ISO 80000-2 supports symbols and notation. Numerical checks and their limits are documented separately in the Circle v1.1.2 Evidence and capability contract; the Trust summary reports the applicable verification and production state.
For a reproducible calculation, share the input link. To cite the tool itself, use the clean Arc calculator URL. Content updates, numerical verification, and domain reviews have separate dates and scopes.
Which numerical range is supported?
All nonzero outputs must be finite normal binary64 values (magnitude at least 2^-1022); exact zero angles are supported. Outside this envelope, NUMERIC_RANGE requests rescaling instead of returning zero, Infinity, or NaN. Results use 12 significant digits. Radius and arc length share the chosen length unit; areas use its square. Signed degree angles and multiple turns are preserved, while the diagram shows at most one turn.
How are request numbers and versions interpreted?
REST and MCP execute_calculation reject a nonzero raw JSON number such as 1e-400 with NUMERIC_UNDERFLOW before it becomes a zero calculation. Literal zero remains valid where the model permits it. Decimal strings preserve input text where declared, but do not extend binary64 precision; a value already sent as 0 cannot be recovered.
Every supplied calculation-version pin must agree. Conflicting header/body pins return VERSION_PIN_CONFLICT (HTTP 409); the body does not override the header. See the request contract for accepted pin locations, input-source limits, and API contract 1.0.1. Calculation-model version and request-contract version are separate.
Related tools
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.
Frequently asked questions
Key distinctions behind the calculation.
How do I calculate arc length in degrees?
Multiply the radius by the central angle and π/180: s = r × θ_deg × π/180. For r = 2 cm and θ = 180°, the arc length is 2π cm.
Does the arc length formula use degrees or radians?
The formula s = rθ uses radians. This calculator accepts degrees and converts them to radians before calculating. A 180° angle is π radians.
Is arc length the same as chord length?
No. Arc length follows the curved edge of the circle; a chord is the straight line between its endpoints. For an angle of 180°, the arc is πr long and the chord is 2r long.
Which units should I use for the radius?
Use any consistent length unit. A radius entered in centimeters gives arc length in centimeters and sector area in square centimeters. The calculator does not convert length units.
What happens at 0° and 360°?
At 0°, the arc length is zero. At 360°, it equals the full circumference, 2πr. A 180° arc is a semicircle and a 90° arc is a quarter circle.
Can I enter a negative angle or more than 360°?
Yes. The calculation preserves the signed angle and full number of turns. A negative angle produces a signed result; 720° gives twice the circumference. The diagram shows at most one turn. Use a nonnegative angle for physical arc distance.
Where does this run?
Interactive calculations run locally in your browser. A shared link sends its URL parameters to the server when opened; REST and MCP requests are calculated on the server.