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Arc Length Calculator

Calculate the length of a circular arc given the radius and central angle in degrees.

Reviewed by Chase FloiedUpdated

This free online arc length calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

The angle subtended by the arc at the center

Results

Arc Length

7.854 units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Arc Length Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Arc Length Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Arc Length Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Arc Length Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Arc Length Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the length of a circular arc given the radius and central angle in degrees. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Arc Length Calculator

An arc is a portion of the circumference of a circle. The arc length is the distance along the curved line between two points on the circle. The formula s = rθ, where θ is in radians, is one of the most elegant in geometry because of its simplicity. This relationship is actually the defining property of radian measure: one radian is the angle that subtends an arc equal in length to the radius. Arc length calculations are essential in engineering for designing curved structures such as roads, bridges, and tracks. They also appear in physics when calculating the distance traveled along circular paths, in cartography for measuring distances along the Earth's surface, and in computer graphics for rendering smooth curves. This calculator converts degrees to radians automatically and computes the arc length for any radius and angle.

The Math Behind It

The formula s = rθ follows directly from the definition of radians. A full circle has angle 2π radians and circumference 2πr, so the arc length for angle θ is (θ/2π) × 2πr = rθ. In degrees, this becomes s = (πrθ°)/180. Arc length is fundamental to differential geometry, where the arc length of any curve y = f(x) from a to b is given by s = ∫ₐᵇ √(1 + (dy/dx)²) dx. For parametric curves x(t), y(t), the formula is s = ∫ √((dx/dt)² + (dy/dt)²) dt. The concept of arc length parameterization, where a curve is parameterized so that the parameter equals the distance traveled along the curve, is crucial in physics (e.g., the Frenet-Serret formulas) and computer-aided design. On the surface of the Earth, arc length along a great circle gives the shortest distance between two points, used in navigation and aviation.

Formula Reference

Arc Length

s = rθ

Variables: r = radius, θ = central angle in radians

Worked Examples

Example 1: Curved road

A road curves along a 200 m radius circle for 30 degrees.

Step 1:θ = 30 × π/180 = π/6 ≈ 0.5236 radians
Step 2:s = 200 × 0.5236

s ≈ 104.72 meters

Common Mistakes & Tips

  • !Forgetting to convert degrees to radians before using s = rθ.
  • !Confusing arc length (a distance) with the central angle (a measure of rotation).
  • !Using the diameter instead of the radius in the formula.

Related Concepts

Sector Area

The area enclosed by an arc and two radii.

Circumference

The arc length of a full circle (360°).

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Circumference Calculator

Sector Area Calculator

Frequently Asked Questions

What is a radian?

A radian is the angle subtended at the center of a circle by an arc whose length equals the radius. One full revolution = 2π radians = 360°.

How do I convert degrees to radians?

Multiply the degree value by π/180. For example, 90° = 90 × π/180 = π/2 radians.