Arc Length Calculator

Calculate the length of a circular arc from radius and angle

Enter the radius and central angle to calculate the arc length. Works with both degrees and radians using the formula: Arc Length = radius × angle (in radians).

Examples

  • Radius = 5, Angle = 90° → Arc Length = 5π/2 ≈ 7.854
  • Radius = 10, Angle = π/3 rad → Arc Length = 10π/3 ≈ 10.472
  • Radius = 3, Angle = 180° → Arc Length = 3π ≈ 9.425
  • Radius = 8, Angle = 1 rad → Arc Length = 8

Important Note

Arc length is always measured in the same units as the radius. The formula Arc Length = r × θ requires the angle in radians.

Other Titles
Understanding Arc Length Calculator: A Comprehensive Guide
Explore arc length calculations, their applications in geometry, engineering, and real-world problem solving

Understanding Arc Length Calculator: A Comprehensive Guide

  • Arc length relates radius, angle, and curved distance on circles
  • The formula connects linear and angular measurements
  • Arc length calculations appear in many practical applications
Arc length is the distance along the curved edge of a circle between two points. This measurement is fundamental in geometry and has numerous practical applications in engineering, construction, and design.
The arc length formula, s = rθ (where s is arc length, r is radius, and θ is the central angle in radians), directly relates the linear distance along a curve to the radius and angular measurement.
Understanding arc length helps solve problems involving circular motion, gear design, architectural features, and any situation where you need to measure distances along curved paths.
Our calculator handles the angle conversion automatically, accepting both degrees and radians while ensuring accurate results.

Basic Arc Length Calculations

  • Quarter circle: r = 4, θ = 90° → arc length = π ≈ 3.142
  • Semicircle: r = 6, θ = π rad → arc length = 6π ≈ 18.850
  • Full circle: r = 2, θ = 360° → arc length = 4π ≈ 12.566

Step-by-Step Guide to Using the Arc Length Calculator

  • Learn how to input radius and angle correctly
  • Understand the relationship between units
  • Master interpreting and applying results
Our arc length calculator simplifies the process of finding curved distances on circles with straightforward inputs.
Step 1: Enter the Radius
Input the radius of your circle. This can be in any unit (inches, centimeters, meters, etc.) - the arc length result will be in the same unit.
Step 2: Input the Central Angle
Enter the central angle that subtends (spans) the arc you want to measure. This is the angle at the center of the circle between the two radii to the arc endpoints.
Step 3: Select Angle Unit
Choose whether your angle is in degrees (more common in everyday use) or radians (used in advanced mathematics).
Step 4: Calculate and Apply
The calculator provides the precise arc length, which you can use directly in your applications or further calculations.

Calculator Usage Examples

  • Pizza slice: r = 12 cm, θ = 30° → arc length = 2π cm ≈ 6.28 cm
  • Clock hand: r = 5 in, θ = 45° → arc length = 5π/4 in ≈ 3.93 in
  • Pipe bend: r = 2 m, θ = π/6 rad → arc length = π/3 m ≈ 1.047 m

Real-World Applications of Arc Length Calculator Calculations

  • Construction: Road curves and architectural elements
  • Manufacturing: Pipe bending and gear design
  • Engineering: Belt drives and pulley systems
  • Navigation: Flight paths and satellite orbits
Arc length calculations are essential across numerous practical applications:
Construction and Architecture:
  • Road Design: Calculating curve lengths for highway design and material estimation.
  • Architectural Features: Determining material needs for arched windows, doorways, and decorative elements.
Manufacturing and Engineering:
  • Pipe Bending: Calculating bend lengths for plumbing and industrial piping systems.
  • Belt Drives: Determining belt lengths for pulleys and drive systems in machinery.
Transportation and Navigation:
  • Aircraft Navigation: Calculating great circle distances for flight planning.
  • Satellite Orbits: Determining orbital path lengths for communication and GPS satellites.

Practical Applications

  • Highway curve: r = 500 m, θ = 30° → curve length = 500π/6 ≈ 261.8 m
  • Garden sprinkler: r = 10 ft, θ = 120° → watered arc = 20π/3 ≈ 20.9 ft
  • Gear tooth: r = 2 cm, θ = 15° → tooth arc = π/6 ≈ 0.524 cm
  • Running track curve: r = 40 m, θ = 180° → curve length = 40π ≈ 125.7 m

Common Misconceptions and Correct Methods in Arc Length Calculations

  • Addressing frequent errors in formula application
  • Clarifying the difference between arc length and chord length
  • Explaining angle unit requirements
Understanding common misconceptions about arc length helps ensure accurate calculations and proper application of results.
Misconception 1: Arc Length Equals Chord Length
Wrong: Confusing the curved arc length with the straight-line chord length between endpoints.
Correct: Arc length follows the curve and is always longer than the chord (except for very small angles). Use s = rθ for arc length, not straight-line distance.
Misconception 2: Using Degrees Directly in the Formula
Wrong: Applying s = rθ with angle in degrees without conversion.
Correct: The formula s = rθ requires the angle in radians. Convert degrees to radians by multiplying by π/180.
Misconception 3: Proportional Scaling Without Proper Formula
Wrong: Assuming arc length is simply proportional to angle in degrees.
Correct: While proportionality exists, use the proper conversion factor. For degrees: s = r × θ × π/180.

Common Error Corrections

  • Wrong: 90° angle with r = 1 gives arc length = 90
  • Correct: 90° = π/2 rad, so arc length = 1 × π/2 ≈ 1.571
  • Wrong: Arc length from A to B equals straight distance AB
  • Correct: Arc length > chord length (except for infinitesimal angles)

Mathematical Derivation and Examples

  • Understanding the geometric basis of arc length
  • Deriving the formula from circle circumference
  • Connecting to angular velocity and rotational motion
The arc length formula derives from the fundamental relationship between a circle's circumference and the angles that create arcs.
Formula Derivation:
Starting with a full circle: circumference = 2πr and central angle = 2π radians
For any arc with central angle θ: arc length/circumference = θ/2π
Therefore: arc length = (θ/2π) × 2πr = rθ
Radian Definition Connection:
By definition, 1 radian is the angle subtending an arc length equal to the radius. This makes s = rθ a natural relationship.
Applications in Physics:
Linear velocity v = rω (where ω is angular velocity) comes directly from the arc length relationship, showing distance traveled per unit time on a circle.

Mathematical Foundations

  • Verification: Full circle with r = 3 → θ = 2π → s = 3(2π) = 6π ✓
  • Unit circle: r = 1, θ = 1 rad → s = 1 (definition of radian) ✓
  • Proportional check: Double angle → double arc length for same radius ✓
  • Physics application: Wheel rotation - distance = radius × angle (radians)