Arc Length Calculator
Calculate Arc Length
Find the length of an arc using radius and central angle. Supports degrees and radians with visual representation.
Arc Length Result
15.708
Step-by-Step Calculation:
Circle Properties:
Circle Visualization:
The arc length is the distance along the curved line making up the arc.
What is Arc Length?
Arc length is the distance measured along the curved line that makes up an arc of a circle. It represents a portion of the circle's circumference and is calculated using the radius and the central angle subtended by the arc. The formula for arc length is fundamental in geometry, trigonometry, and many practical applications involving circular motion and curved paths.
Arc Length Formulas
Using Degrees
Angle in degrees
Common in practical applications
Using Radians
Angle in radians
Simplest formula
Using Chord Length
When chord length known
For partial arcs
Using Sector Area
When sector area known
A = sector area
Arc Length Formulas in Detail
1. Basic Arc Length Formula
When the central angle θ is in radians:
Arc Length = r × θ
Where:
• r = radius of the circle
• θ = central angle in radians
• Result in same units as radius
2. Arc Length with Degrees
When the central angle θ is in degrees, convert to radians first:
Arc Length = (θ × π × r) / 180
Or:
Arc Length = r × (θ × π/180)
Conversion factor: 1° = π/180 radians ≈ 0.0174533 rad
3. Relationship with Circumference
The arc length is a fraction of the full circumference:
Arc Length = (θ/360°) × 2πr (for degrees)
Arc Length = (θ/2π) × 2πr = rθ (for radians)
Full circle: θ = 360° or 2π rad → Arc = 2πr (circumference)
Real-World Applications
Engineering & Construction
- Road design: Calculating curved road sections and banking angles
- Bridge construction: Designing arch bridges and curved supports
- Railway tracks: Determining track curvature and superelevation
- Pipe bending: Calculating material length for curved pipes
Manufacturing & Design
- Metal fabrication: Determining material length for curved components
- Textile industry: Calculating fabric for curved patterns
- Automotive design: Wheel alignment and turning radius calculations
- Product design: Rounded corners and curved edges
Astronomy & Navigation
- Orbital mechanics: Calculating satellite orbits and trajectories
- Celestial navigation: Measuring angular distances between stars
- GPS systems: Great circle distances on Earth's surface
- Telescope design: Field of view calculations
Sports & Recreation
- Track & field: Lane measurements on curved tracks
- Swimming pools: Lane lengths in curved pools
- Amusement rides: Roller coaster track design
- Sports fields: Curved boundary markings
Common Arc Length Examples
| Description | Radius | Angle | Arc Length | Application |
|---|---|---|---|---|
| Quarter Circle | 10 units | 90° | 15.708 units | 90° corner rounding |
| Semicircle | 5 units | 180° | 15.708 units | Half-round arch |
| One-third Circle | 6 units | 120° | 12.566 units | Tricycle wheel rotation |
| Small Arc | 100 units | 30° | 52.36 units | Road curve |
| Full Circle | 7 units | 360° | 43.982 units | Complete circumference |
| Clock Minute Hand | 5 cm | 6° (1 min) | 0.5236 cm | Clock mechanics |
Related Circle Measurements
| Measurement | Formula | Relationship to Arc | Example |
|---|---|---|---|
| Circumference | C = 2πr | Arc for θ = 360° | r=10 → C=62.832 |
| Sector Area | A = (θ/360°) × πr² | A = (L × r)/2 | θ=90°, r=10 → A=78.54 |
| Chord Length | c = 2r × sin(θ/2) | Straight line distance | θ=90°, r=10 → c=14.142 |
| Segment Height | h = r × (1 - cos(θ/2)) | Max distance from chord to arc | θ=90°, r=10 → h=2.9289 |
Step-by-Step Calculation Process
Example 1: Calculate arc length for r=10, θ=90°
- Given: Radius r = 10 units, Central angle θ = 90°
- Convert degrees to radians: θ_rad = 90° × (π/180) = π/2 ≈ 1.5708 rad
- Apply arc length formula: L = r × θ_rad
- Calculate: L = 10 × 1.5708 = 15.708 units
- Alternative using degrees formula: L = (90 × π × 10) / 180 = 15.708 units
- Verify: Full circumference = 2πr = 62.832 units, Arc ratio = 90°/360° = 0.25, 0.25 × 62.832 = 15.708
Example 2: Calculate arc length for r=5, θ=2.5 radians
- Given: Radius r = 5 units, Central angle θ = 2.5 rad
- Since angle is already in radians, use direct formula: L = r × θ
- Calculate: L = 5 × 2.5 = 12.5 units
- Convert to degrees: 2.5 rad × (180/π) ≈ 143.24°
- Full circumference = 2π × 5 = 31.416 units
- Arc ratio = 2.5/(2π) ≈ 0.3979, 0.3979 × 31.416 ≈ 12.5 units
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Frequently Asked Questions (FAQs)
Q: What's the difference between arc length and chord length?
A: Arc length is the distance along the curved path of the arc. Chord length is the straight-line distance between the arc's endpoints. For the same arc, chord length is always shorter than arc length (except for very small angles where they're approximately equal).
Q: How do I calculate arc length without knowing the angle?
A: If you know the chord length (c) and radius (r), use: θ = 2 × sin⁻¹(c/(2r)), then L = r × θ. If you know sector area (A) and radius: L = (2 × A) / r.
Q: Can arc length be greater than circumference?
A: No, arc length is always less than or equal to circumference. For angles greater than 360°, the arc wraps around the circle multiple times, but the actual path length is still (θ mod 360°) fraction of circumference.
Q: How is arc length used in real life?
A: Arc length calculations are used in road design (curves), architecture (arches), manufacturing (bending materials), sports (track lanes), astronomy (planetary orbits), and many engineering applications involving circular motion or curved surfaces.
Master arc length calculations with Toolivaa's free Arc Length Calculator, and explore more geometry tools in our Geometry Calculators collection.