The Arc Length Calculator computes the distance along a circle’s curve from a radius and a central angle. Enter either degrees or radians, and the calculator returns arc length plus helpful intermediate values.
What Is Arc Length?
Arc length is the distance measured along the curved edge of a circle. It’s used in geometry, engineering, and design when you need the length of a curved path, not the straight-line distance.
Arc length depends on two things:
- Radius (r): how far the arc is from the center.
- Central angle (θ): the angle that “cuts out” the arc at the circle’s center.
Arc Length Formula (Core Concept)
The standard formula is:
s = r · θ
where:
- s = arc length
- r = radius
- θ = central angle in radians
If your angle is in degrees, convert it first:
θ(radians) = θ(degrees) · π / 180
Substituting into the main equation gives the degrees form:
s = (π · r · θ°) / 180
How the Calculator Works
This Arc Length Calculator uses the correct formula based on your selected angle unit. It then computes:
- Arc length (s): the final distance along the curve.
- Angle in radians: when you input degrees, it converts to radians for the main equation.
Internally, the calculator applies:
| Input | Conversion/Formula |
|---|---|
| Angle in degrees | θ = θ° · π / 180, then s = r · θ |
| Angle in radians | s = r · θ |
Units and What They Mean
Arc length has the same length units as the radius. For example, if the radius is in meters, arc length is in meters.
Common radius units include:
- mm (millimeters)
- cm (centimeters)
- m (meters)
- in (inches)
- ft (feet)
The calculator lets you choose the radius unit so the output matches your measurement system.
Step-by-Step: How to Use an Arc Length Calculator
- Enter the radius (r): the distance from the center of the circle to the curve.
- Enter the central angle (θ): the angle that spans the arc.
- Select the angle unit: degrees or radians.
- Select the radius unit: mm, cm, m, in, or ft.
- Read the result: arc length in the chosen radius unit.
If inputs are missing or invalid, the calculator will highlight the field and prompt you to correct it.
Practical Examples (Real Life Use Cases)
Example 1: Designing a Curved Rail
Suppose you’re installing a curved rail that follows a circle. The rail’s path has a radius of 2.5 m and spans a central angle of 60°. You need the length of that curved segment.
Using the formula for degrees: s = (π · r · θ°) / 180.
- r = 2.5 m
- θ° = 60°
The arc length is the distance you cut and install for the curved section.
Example 2: Measuring a Belt Segment
A belt wraps around a pulley. If the pulley radius is 4 in and the belt covers 1.2 radians, the belt segment length is exactly the arc length.
- r = 4 in
- θ = 1.2 rad
Because the angle is already in radians, use s = r · θ. This is common in mechanical and robotics calculations where angles are measured in radians.
Common Mistakes to Avoid
- Using degrees in the radians formula: if your calculator expects radians, always convert degrees to radians.
- Mixing units: if you enter radius in centimeters but expect meters, convert first or select the correct unit in the calculator.
- Confusing arc length with chord length: arc length follows the curve; chord length is the straight line between arc endpoints.
Frequently Asked Questions
How do I find arc length when my angle is in degrees?
Use the degrees form of the formula: s = (π · r · θ°) / 180. First confirm your radius r is in the units you want for the answer. Then plug in θ° and compute. This matches the radian method after converting degrees to radians.
What is the difference between arc length and chord length?
Arc length measures the distance along the curved edge of the circle. Chord length measures the straight-line distance between two points on the circle. For the same radius and angle, arc length is always larger than chord length except at 0°.
Why must the angle be in radians for s = r · θ?
The formula s = r · θ is derived from the geometry of circles where θ is measured as a fraction of a full rotation. That fraction works directly when θ is in radians. If you use degrees, the scale factor is wrong unless you convert.
Can the arc length be negative?
In most geometry and measurement contexts, arc length is treated as a positive distance. A negative angle usually indicates direction (clockwise vs. counterclockwise), not physical “negative distance.” If you enter a negative angle, the calculator will return a negative value.
Is arc length always less than the circle’s circumference?
Yes, as long as your central angle is between 0° and 360° (or 0 and 2π radians). For angles in that range, arc length is a portion of the circumference. At 360° (2π), arc length equals the full circumference.
Quick Reference: Formula Summary
- Radians: s = r · θ
- Degrees: s = (π · r · θ°) / 180
- Conversion: θ(rad) = θ° · π / 180
Use this Arc Length Calculator any time you need the distance along a curve—whether you’re working on math homework, CAD models, or real-world curved installations.
