Quick answer
Arc length L = r × θ with θ in radians, or 2πr × θ/360° for degrees. A 90° arc of a circle with radius 6 has length 9.4248.
Enter the radius and central angle in degrees. The calculator returns the arc length.
Related tools: Sector Area Calculator: Area and arc length of a circular sector. Radius Calculator: Radius of a circle from diameter, circumference or area. Circle Diameter Calculator: Diameter of a circle from radius, circumference or area.
How to use the Arc Length Calculator
- Enter radius (above 0).
- Enter central angle (°) (between 0 and 360).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Radius | A number | > 0 |
| Central angle | A number, in ° | > 0, ≤ 360 |
Formula and method
Formula
L = r × θ (θ in radians) = 2πr × θ/360°
IEEE-754 double precision; shown to at most 6 decimal places.
Fraction of the circumference.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Radius = 6; Central angle = 90.
| Result | Value |
|---|---|
| Arc length | 9.424778 |
Example 2
Inputs: Radius = 1; Central angle = 360.
| Result | Value |
|---|---|
| Arc length | 6.283185 |
The first line is the headline result (arc length). IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
90° is a quarter of the circle. The full circumference is 2π × 6 = 37.699, a quarter of that is 9.4248. In radians: 6 × π/2 = 9.4248.
Related ideas
Sector area = ½ r² θ and chord length = 2r sin(θ/2). The angle can be up to 360°. A 180° arc gives half the circumference.
How the result changes with radius
Holding every other input at the first worked example, this table changes only radius. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Radius | Arc length |
|---|---|
| 3 | 4.712389 |
| 6 | 9.424778 |
| 12 | 18.849556 |
| 24 | 37.699112 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Central angle: entering "400" shows "Central angle must be at most 360." beside that field and no result.
Choosing between the Arc Length Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Sector Area Calculator](/math/sector-area/) | radius and central angle | Sector area |
| [Radius Calculator](/math/radius-calculator/) | i know the and value | Radius |
| [Circle Diameter Calculator](/math/circle-diameter/) | i know the and value | Diameter |
| [Circumference Calculator](/math/circumference/) | i know the and value | Circumference |
| [Chord Length Calculator](/math/chord-length/) | radius and central angle | Chord length |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Angle up to 360°
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for central angle to confirm this.
Frequently asked questions
How do I convert degrees to radians?
Multiply degrees by π/180, so 90° is π/2 = 1.5708 radians.
What is arc length at 360°?
The full circumference.
What is a sector?
The region bounded by an arc and two radii.
What does the Arc Length Calculator calculate?
Length of a circular arc. It returns Arc length.
What formula does the Arc Length Calculator use?
L = r × θ (θ in radians) = 2πr × θ/360°. Fraction of the circumference.
What do I need to enter in the arc Length Calculator?
You enter Radius and Central angle (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Angle up to 360°.
What is an example of a arc Length result?
With radius 6 and central angle 90, the arc length is 9.424778.
Related calculators
- Sector Area Calculator — Area and arc length of a circular sector.
- Radius Calculator — Radius of a circle from diameter, circumference or area.
- Circle Diameter Calculator — Diameter of a circle from radius, circumference or area.
- Circumference Calculator — Circumference of a circle.
- Chord Length Calculator — Length of a chord from the radius and central angle.
- Diagonal Length Calculator — Diagonal of a rectangle or the space diagonal of a box.
- Area Calculator — Area of common two-dimensional shapes.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Rhombus Calculator — Area, side and perimeter of a rhombus from its diagonals.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.