Arc Length Calculator
Calculate the length of a circular arc using radius and central angle. Support for both degrees and radians with step-by-step solutions.
Arc Length Inputs
Common Angle Conversions
30°π/6 ≈ 0.524
45°π/4 ≈ 0.785
60°π/3 ≈ 1.047
90°π/2 ≈ 1.571
120°2π/3 ≈ 2.094
180°π ≈ 3.142
270°3π/2 ≈ 4.712
360°2π ≈ 6.283
Arc Length Properties
Arc Length Formula: s = rθ (θ in radians)
Degree Formula: s = (θ/360°) × 2πr
Radian Conversion: radians = degrees × (π/180)
Full Circle: 360° = 2π radians
Understanding Arc Length
Arc length is a fundamental concept in circle geometry that measures the distance along a circular path between two points. Unlike chord length (straight-line distance), arc length follows the curve of the circle.
Key Concepts:
- Central Angle: The angle formed at the center of the circle by two radii
- Radius: The distance from the center to any point on the circle
- Radian: The natural unit for measuring angles (2π radians = 360°)
- Arc Sector: The pie-slice shaped region bounded by the arc and two radii
Formula Derivation:
The arc length formula s = rθ comes from the relationship between the full circumference (2πr) and the central angle. Since 2π radians represents a full circle, any angle θ represents the fraction θ/(2π) of the full circle.
Real-World Applications:
- Designing curved roads and railway tracks
- Calculating distances along Earth's surface (great circle navigation)
- Engineering curved bridges and architectural arches
- Robotics and mechanical arm movement calculations
- Computer graphics and animation path calculations
Frequently Asked Questions - Arc Length Calculator
Arc length is the distance along the curved line of a circle's circumference between two points. It's measured along the curve, not as a straight line between the endpoints.