📐 Arc Length Calculator
Calculate the length of an arc on a circle
How to Use This Calculator
Enter the Radius
Input the radius of the circle. This is the distance from the center to any point on the circle.
Select Angle Unit
Choose whether your angle is in degrees or radians. Most commonly, angles are in degrees.
Enter the Central Angle
Input the central angle that subtends the arc. For degrees, use values between 0 and 360.
Calculate
Click the "Calculate Arc Length" button to get the result.
Formula
Arc Length = r × θ
Where θ must be in radians
Where:
- r = radius of the circle
- θ = central angle in radians
- Arc Length = length along the curve
Converting Degrees to Radians:
θ (radians) = θ (degrees) × (π / 180)
Example 1: Find arc length for r = 5 units, θ = 90°
Step 1: Convert to radians: 90° × (π/180) = π/2 ≈ 1.5708 rad
Step 2: Calculate: Arc Length = 5 × 1.5708 = 7.854 units
This is 1/4 of the circumference (90° = 360°/4)
Example 2: Find arc length for r = 10 units, θ = 2.5 radians
Arc Length = 10 × 2.5 = 25 units
No conversion needed since angle is already in radians.
Example 3: Find arc length for r = 3 units, θ = 60°
Step 1: Convert: 60° × (π/180) = π/3 ≈ 1.0472 rad
Step 2: Calculate: Arc Length = 3 × 1.0472 = 3.1416 units
Note: This equals π, since 60° is 1/6 of a full circle.
About Arc Length Calculator
The Arc Length Calculator finds the distance along the curved edge of a circle (arc) between two points. It's calculated by multiplying the radius by the central angle (in radians) that the arc subtends.
When to Use This Calculator
- Geometry: Calculate arc lengths in circles and sectors
- Engineering: Design curved structures, roads, and tracks
- Construction: Determine material needed for curved paths
- Architecture: Plan circular features and curved walls
- Physics: Analyze circular motion and trajectories
- Education: Learn and practice arc length concepts
Why Use Our Calculator?
- ✅ Instant Results: Calculate arc length immediately
- ✅ Flexible Units: Works with degrees or radians
- ✅ Accurate: Precise mathematical calculations
- ✅ Educational: Shows conversion steps and formulas
- ✅ 100% Free: No registration required
- ✅ Mobile Friendly: Works on all devices
Understanding Arc Length
Arc length is the distance along the curved part of a circle. It's different from a straight-line chord connecting the same two points. The formula uses radians because it simplifies the mathematics.
- Full circle: Arc length = 2πr (when θ = 360° or 2π radians)
- Semicircle: Arc length = πr (when θ = 180° or π radians)
- Quarter circle: Arc length = πr/2 (when θ = 90° or π/2 radians)
- The arc length is proportional to both radius and angle
Real-World Applications
Road Design: Calculate the length of curved road segments for construction and material estimation.
Architecture: Determine the length of curved windows, doorways, or decorative elements.
Engineering: Calculate cable length for suspension bridges or track length for circular railways.
Sports: Measure running track curves, racing lines, or curved paths in stadiums.
Frequently Asked Questions
What is arc length?
Arc length is the distance along the curved part of a circle between two points. It's measured along the circle's circumference, not as a straight line.
Why do we need to convert degrees to radians?
The formula Arc Length = r × θ works when θ is in radians. If you have degrees, convert using: radians = degrees × (π/180). Our calculator does this automatically!
What's the difference between arc length and chord length?
Arc length is the distance along the curved path. Chord length is the straight-line distance between the two endpoints. For small angles, they're similar, but for larger angles, the arc is longer.
How do I find arc length if I know the chord length?
You'd need additional information like the radius or central angle. Use the chord length formula with radius to find the angle, then use arc length = r × θ.
Can arc length be greater than the circumference?
No, arc length is always ≤ 2πr (the full circumference). The maximum occurs when the angle is 360° (2π radians), giving a complete circle.
How is arc length related to sector area?
Both depend on the central angle. Sector Area = (1/2) × r² × θ (in radians). Arc Length = r × θ. So Area = (1/2) × r × Arc Length.
What if the angle is in revolutions instead of degrees?
Convert revolutions to radians: 1 revolution = 360° = 2π radians. For example, 0.5 revolutions = π radians, so arc length = r × π.