Arc Length Calculator

Arc Length Calculator – Calculate Arc Length of Circle & Sector Online

  • Adam
  • November 22, 2025

Free arc length calculator to find the length of an arc using radius, angle, or chord. Calculate arc length in degrees and radians with formulas and step-by-step solutions.

Arc Length Calculator

Calculate Arc Length of Circle & Sector Instantly

Calculate Arc Length

What is Arc Length?

Arc length is the distance along the curved line that makes up the arc. It represents a portion of the circumference of a circle, measured along the curve from one endpoint to the other. Understanding arc length is fundamental in geometry, trigonometry, and calculus for calculating distances along curved paths.

Key Components of Arc Length

  • Radius (r): The distance from the center of the circle to any point on the arc
  • Central Angle (θ): The angle subtended by the arc at the center of the circle
  • Arc Length (L): The distance along the curve of the arc
  • Chord: The straight line connecting the two endpoints of the arc
  • Sector: The pie-shaped region bounded by two radii and the arc
  • Segment: The region between the chord and the arc

Real-World Applications

Arc length calculations are used extensively in:

  • Architecture and construction for curved structures and arches
  • Engineering for designing gears, pulleys, and curved roads
  • Navigation and GPS for calculating distances along curved paths
  • Astronomy for measuring angular distances between celestial objects
  • Sports for analyzing curved ball trajectories and running track distances

Arc Length Formulas

The formula for calculating arc length depends on whether the central angle is measured in degrees or radians. Both formulas are derived from the relationship between the arc and the circle's circumference.

Primary Arc Length Formulas

Arc Length (Radians)

Formula:

L = rθ

Simplest form when angle θ is in radians. Multiply radius by angle directly.

Arc Length (Degrees)

Formula:

L = (θ/360) × 2πr

When angle θ is in degrees. Take fraction of full circumference.

Alternative Formula (Degrees)

Formula:

L = (πrθ) / 180

Simplified version of the degree formula, mathematically equivalent

Related Formulas

Sector Area

A = (1/2) × r² × θ (radians)

A = (θ/360) × πr² (degrees)

Chord Length

c = 2r × sin(θ/2)

Straight line distance between arc endpoints

Sector Perimeter

P = 2r + L

Sum of two radii plus arc length

Detailed Example: Calculate Arc Length

Find the arc length of a circle with radius 10 cm and central angle of 60°:

Given Information:

  • Radius (r) = 10 cm
  • Central Angle (θ) = 60°
  • Need to find: Arc Length (L)

Solution Steps:

Step 1: Use arc length formula for degrees

L = (θ/360) × 2πr

Step 2: Substitute values

L = (60/360) × 2π(10)

L = (1/6) × 20π

Step 3: Calculate

L = (20π)/6 = 10π/3

L ≈ 10.47 cm

Result: The arc length is approximately 10.47 cm. This represents 1/6 of the circle's circumference (since 60° is 1/6 of 360°).

Degrees vs Radians in Arc Length

Understanding the difference between degrees and radians is crucial for arc length calculations. Both measure angles, but they use different scales and require different formulas.

Using Degrees

Most common in everyday measurements:

  • Full circle = 360°
  • Half circle = 180°
  • Quarter circle = 90°
  • Formula: L = (θ/360) × 2πr

Using Radians

Preferred in mathematics and calculus:

  • Full circle = 2π rad ≈ 6.28 rad
  • Half circle = π rad ≈ 3.14 rad
  • Quarter circle = π/2 rad ≈ 1.57 rad
  • Formula: L = rθ (simplest!)

Conversion Formulas

Degrees to Radians:

radians = degrees × (π/180)

Radians to Degrees:

degrees = radians × (180/π)

Common Arc Lengths Reference

Here are commonly calculated arc lengths for standard angles. These serve as quick references and help verify your calculations.

Arc Lengths for Unit Circle (r = 1)

Angle (Degrees) Angle (Radians) Arc Length (r=1) Fraction of Circle
30° π/6 ≈ 0.524 0.524 1/12
45° π/4 ≈ 0.785 0.785 1/8
60° π/3 ≈ 1.047 1.047 1/6
90° π/2 ≈ 1.571 1.571 1/4
180° π ≈ 3.142 3.142 1/2
360° 2π ≈ 6.283 6.283 Full circle

Frequently Asked Questions

What is arc length?

Arc length is the distance along the curved line that makes up the arc. It is a portion of the circumference of a circle. For a circle with radius r and central angle θ, arc length L = rθ (in radians) or L = (θ/360) × 2πr (in degrees). For example, a quarter circle (90°) has an arc length equal to one-fourth of the circumference.

How do you calculate arc length?

To calculate arc length: 1) If angle is in radians, use L = rθ where r is radius and θ is central angle, 2) If angle is in degrees, use L = (θ/360) × 2πr or L = (πrθ)/180. For example, an arc with radius 10 cm and angle 60° has length L = (60/360) × 2π(10) = (1/6) × 20π ≈ 10.47 cm.

What is the formula for arc length?

The arc length formula depends on angle units. For radians: L = rθ (simplest form - multiply radius by angle). For degrees: L = (θ/360) × 2πr or L = (πrθ)/180. Where L is arc length, r is radius, and θ is the central angle. The radian formula is simpler, which is why radians are preferred in mathematics.

How to find arc length without angle?

To find arc length without the angle, you need the chord length c and radius r. First calculate the central angle using: θ = 2 × arcsin(c/2r) in radians. Then use the arc length formula L = rθ. Alternatively, if you know the sector area A and radius r, find the angle first: θ = 2A/r² (radians), then calculate L = rθ.

What is the difference between arc length and circumference?

Circumference is the total distance around the entire circle, calculated as C = 2πr. Arc length is only a portion of the circumference corresponding to a specific central angle. Arc length is a fraction of circumference: L = (θ/360°) × C when θ is in degrees. A full circle (360°) has arc length equal to circumference.

Why use radians instead of degrees for arc length?

Radians simplify the arc length formula to L = rθ, making calculations easier and more intuitive. In radians, the arc length equals radius times angle directly. Radians are also natural units for circular motion - one radian is the angle where arc length equals radius. This simplicity is why calculus, physics, and engineering prefer radians.

How do you find radius from arc length?

To find radius from arc length, rearrange the arc length formula. If angle is in radians: r = L/θ. If angle is in degrees: r = 180L/(πθ) or r = 360L/(2πθ). For example, if arc length is 15 cm and angle is 90° (π/2 radians): r = L/θ = 15/(π/2) = 30/π ≈ 9.55 cm.

What is the relationship between arc length and sector area?

Arc length and sector area are related through the radius. If you know arc length L and radius r, sector area A = (1/2) × L × r. Alternatively, both depend on the central angle: L = rθ and A = (1/2)r²θ (in radians). Therefore, A = (L × r)/2, showing area equals half the product of arc length and radius.

Disclaimer: This arc length calculator provides accurate results based on standard geometric formulas: L = rθ (radians) and L = (θ/360) × 2πr (degrees). Calculations use π ≈ 3.14159. Results are for educational purposes and general calculations. For critical engineering or construction applications, always verify calculations independently and consult with professionals. Ensure consistent units throughout your calculations.

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