What is Arc Length?
Arc length is the distance along the curved edge of a part of a circle. Imagine walking along the rim of a round pizza crust; the distance you walk is the arc length. This concept is fundamental in geometry, engineering, and physics, relating linear distance to rotational motion.
The Formula
s = r × θ
r: Radius | θ: Angle (in radians)
Understanding Radians vs. Degrees
The standard formula requires the angle to be in radians. Why? Because a radian is defined such that the arc length is equal to the radius when the angle is exactly 1 radian.
- Degrees: A full circle is 360°. Formula: s = 2πr × (θ/360).
- Radians: A full circle is 2π radians. Formula: s = rθ.
Real-World Examples
Travel Distance
Scenario: Calculating flight distance along the curved surface of the Earth.
Engineering
Scenario: Determining the length of a belt connecting two pulleys or the spacing of gear teeth.
How the Arc Length Is Computed
The calculator uses s = r × θ, where s is arc length, r is radius, and θ is the central angle in radians. If you enter the angle in degrees, it is converted first: θ_radians = (π/180) × θ_degrees. The result has the same units as the radius (e.g. meters, feet). For a full circle (360° or 2π rad), s = 2πr (the circumference). For a semicircle (180° or π rad), s = πr. The formula assumes a perfect circle; for ellipses or other curves, different formulas apply. This tool is for informational use only; for engineering or construction, verify with standards.
How to Use This Arc Length Calculator
Enter the radius (or diameter) and the central angle of the arc—in degrees or radians, depending on the tool. The Arc Length Calculator uses the formula s = r × θ (with θ in radians) or the equivalent in degrees: s = (π/180) × r × θ°. The result is the length of the arc along the circle. Use it for circular segments, belt lengths between pulleys, curved fences, or any geometry problem involving a portion of a circle. For the full circumference, use 360° (or 2π radians); for a semicircle, 180° (or π).
When Arc Length Is Needed
Arc length appears in engineering (belt drives, cam profiles), architecture (curved walls, arches), land surveying, and math education. Ensure units are consistent (e.g. radius and result in the same length unit). This calculator assumes a circular arc; for other curves, integration or specialized formulas are required. For sector area or segment area, use an area calculator that supports circular segments.
The formula s = r × θ (with θ in radians) is exact for any circular arc. If the angle is in degrees, convert first: θ_radians = (π/180) × θ_degrees. The full circle has circumference 2πr; a semicircle has length πr. For belt length between two pulleys, you often need the arc lengths on both wheels plus the straight tangent segments; this tool gives one arc. Keep radius and angle in consistent units. This tool is for informational use only; for critical design verify with engineering standards.
Tips for Use
Use the same units for radius and result (e.g. both in meters). For belt length between two pulleys, you need the arc on each pulley plus the straight tangent segments; this calculator gives one arc. For sector area (pie slice), use area = (θ/2) × r² in radians or (θ/360) × πr² in degrees. For chord length (straight line between arc endpoints), use chord = 2r × sin(θ/2). Keep angle in the correct unit (degrees or radians) as selected. This tool is for circular arcs only; for other curves, use calculus or specialized formulas. In physics, arc length relates to angular displacement: s = rθ is used in rotational motion (e.g. distance traveled by a point on a rotating wheel). In surveying, arc length along the Earth's surface (with Earth radius) approximates great-circle distance for short arcs. For critical design or construction, verify formulas and units with engineering standards. Ensure the angle unit (degrees or radians) matches the calculator setting for correct results.
Limitations
This calculator assumes a circular arc with a given radius and central angle. It does not handle ellipses, parabolas, or arbitrary curves. For very large or very small numbers, rounding and floating-point precision may affect the result. Units (meters, feet, etc.) are not converted; use the same unit for radius and interpret the result in that unit. This tool is for informational use only; for construction or design, verify with qualified professionals.
Disclaimer
This tool is for informational use only. For critical design or construction, verify formulas and units with engineering standards.
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About the Arc Length
A precise module that calculates the true length of circle arcs in both radians and degrees.
Arc Length How to Use?
- Enter the radius of the circle
- Enter the central angle
- Select the angle unit (deg/rad)
- See the result along with the solution steps
Frequently Asked Questions
How is arc length calculated?
Arc length (s) is calculated by the product of the radius (r) and the central angle (θ) in radians: s = r × θ. If the angle is in degrees, multiply by π/180 first.
How are degrees converted to radians?
To convert an angle in degrees to radians, you need to multiply the angle by π/180. For example, 180° = π radians.
What is the difference between Arc Length and Chord Length?
Arc length is the distance along the curved edge of the circle. Chord length is the straight-line distance between the two endpoints of the arc.
Can I use the diameter instead of the radius?
Yes, simply divide the diameter by 2 to get the radius, then use the standard formula. Or use s = (θ/360) × π × d for degrees.
Arc Length Features
- Automatic Degree-Radian conversion
- Precise usage of Pi
- Geometric visualization support
- Mobile-friendly design