Arc Length Calculator
Calculate arc length, sector area, and circle properties using various input methods
Calculate Arc Length
Most common method using radius and central angle
Circle radius
Central angle
Arc Length Results
Enter positive values for radius and central angle
Example Calculations
Basic Arc Length
Given: Radius = 15 cm, Central Angle = 45° (π/4 radians)
Formula: Arc Length = r × θ
Calculation: L = 15 × π/4 ≈ 11.78 cm
Sector Area: A = r²θ/2 = 15² × π/4 / 2 ≈ 88.36 cm²
Using Chord Length
Given: Chord = 10 cm, Central Angle = 60°
Formula: r = c/(2×sin(θ/2))
Calculation: r = 10/(2×sin(30°)) = 10 cm
Arc Length: L = 10 × π/3 ≈ 10.47 cm
Arc Length Formulas
Basic Formula
L = r × θ (θ in radians)
Most fundamental formula
Using Degrees
L = (θ × π × r) / 180
When angle is in degrees
Sector Area
A = r² × θ / 2
Area of circular sector
From Chord
r = c / (2 × sin(θ/2))
Radius from chord length
Arc Length Tips
Arc length is always proportional to the central angle
Use radians for simpler calculations (L = r × θ)
Full circle arc length equals circumference (2πr)
Semicircle arc length is πr (half circumference)
Arc length is always greater than chord length
Understanding Arc Length and Circle Sectors
What is Arc Length?
Arc length is the distance along the curved line of a circle's circumference between two points. It's proportional to the central angle and the radius of the circle.
Key Relationships:
- Arc length increases with larger radius
- Arc length increases with larger central angle
- Arc length = (angle/360°) × circumference
- Measured in linear units (cm, m, inches, etc.)
- Always longer than the chord length
Calculation Methods
Radius + Angle
L = r × θ (most direct method)
Diameter + Angle
L = (d/2) × θ
Area + Angle
Find radius first: r = √(2A/θ)
Chord + Angle
Find radius: r = c/(2×sin(θ/2))
Arc Length vs. Other Circle Measurements
| Measurement | Formula | Description |
|---|---|---|
| Arc Length | L = r × θ | Distance along the curved path |
| Chord Length | c = 2r × sin(θ/2) | Straight line distance between endpoints |
| Sector Area | A = r² × θ / 2 | Area of the pie-slice region |
| Circumference | C = 2πr | Full circle perimeter (θ = 2π) |