How do I convert degrees to radians?

To convert degrees to radians, multiply the degree value by π/180. For example, 60° = 60 × π/180 = π/3 radians.

Why does arc length increase linearly with radius?

Arc length is directly proportional to radius because L = r × θ. If you double the radius while keeping the angle constant, the arc length also doubles.

Can I use diameter instead of radius?

Yes, you can use diameter. First convert diameter to radius (r = d/2), then use the formula L = r × θ. Alternatively, use L = πd × (θ/360°) when the angle is in degrees.

What is the relationship between arc length and circumference?

For a full circle (360° or 2π radians), the arc length equals the circumference: L = C = 2πr = πd. For partial arcs, the arc length is a fraction of the circumference based on the angle.

Arc Length Calculator

Calculate the arc length of a circle using radius or diameter and the central angle in degrees or radians. This calculator provides step-by-step solutions and multiple formula representations.

Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)

Arc Length Calculator

Input Type

Radius (r)

Angle (θ)

How It Works

The arc length formula in radians is: L = r × θ, where r is the radius and θ is the angle in radians.

When using degrees, convert to radians first: θ_rad = θ_deg × π/180, then use L = r × θ_rad.

Alternative formula using diameter: L = πd × (θ/360°) when angle is in degrees.

The arc length is proportional to both the radius and the angle measure.

For a full circle (360° or 2π radians), the arc length equals the circumference: C = 2πr = πd.

Step-by-step Examples

Example 1: Using Radius and Degrees

Calculate the arc length of a circle with radius 5 units and central angle 60°.

Given: r = 5, θ = 60°
Convert angle to radians: θ = 60° × π/180 = π/3 radians
Apply formula: L = r × θ = 5 × (π/3) = 5π/3 ≈ 5.236 units

Example 2: Using Diameter and Radians

Find the arc length when diameter is 10 units and angle is 1.5 radians.

Given: d = 10, θ = 1.5 radians
Calculate radius: r = d/2 = 10/2 = 5
Apply formula: L = r × θ = 5 × 1.5 = 7.5 units

Common Mistakes

Forgetting to convert degrees to radians before using the formula L = r × θ.
Using the diameter value directly in the formula without converting to radius first.
Confusing arc length with sector area - remember arc length is a linear measurement, not an area.
Not checking if the angle is in the valid range (0° to 360° or 0 to 2π radians).

Use Cases

Engineering: Calculating the length of curved paths in mechanical designs.
Architecture: Determining the arc length of curved structures like arches and bridges.
Physics: Analyzing circular motion and calculating distances traveled along circular paths.
Mathematics: Solving geometry problems involving circles and arcs.
Manufacturing: Measuring curved components and calculating material requirements.

Frequently Asked Questions

Common questions about arc length calculations.

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