Input Parameters

Radius (r)

Angle (θ)

Angle Unit

Degrees Radians

What-If Analysis

Adjust the radius to see how it affects arc length and chord length (with fixed angle)

Fixed Angle (degrees)

Radius Range

1 10 20

Current Radius: 10

Arc Length: 10.47

Chord Length: 10.00

Understanding Arc Length and Chord Calculations

Calculating arc length, chord length, and sector area is essential in many fields including engineering, architecture, and design. These measurements help determine material requirements, structural integrity, and aesthetic proportions in circular designs.

Key Formulas

Arc Length (L): L = r × θ (where θ is in radians)
Chord Length (c): c = 2 × r × sin(θ/2)
Sector Area (A): A = (θ/2) × r² (where θ is in radians)
Segment Area: A = (r²/2) × (θ - sinθ)

Practical Example

Imagine designing a circular archway with a radius of 5 meters and a central angle of 60 degrees:

First convert 60° to radians: 60 × (π/180) ≈ 1.0472 radians
Arc Length = 5 × 1.0472 ≈ 5.236 meters
Chord Length = 2 × 5 × sin(30°) = 10 × 0.5 = 5 meters
Sector Area = (1.0472/2) × 25 ≈ 13.09 square meters

Reference Table

Angle (degrees)	Angle (radians)	Chord Factor (c/r)	Arc Factor (L/r)
30	0.5236	0.5176	0.5236
60	1.0472	1.0000	1.0472
90	1.5708	1.4142	1.5708
120	2.0944	1.7321	2.0944
180	3.1416	2.0000	3.1416

Tips & Warnings

Always check if your angle is in degrees or radians before calculation
For small angles (less than 10°), chord length and arc length are nearly identical
Chord length reaches its maximum at 180° (equal to the diameter)
Be cautious with floating-point precision when working with very small or very large circles

Frequently Asked Questions

What's the difference between chord length and arc length?

Chord length is the straight-line distance between two points on a circle, while arc length is the distance along the curved line between those two points.

Why do I need to convert degrees to radians?

The mathematical formulas for arc length and sector area are derived using radians. While you can use degrees with conversion factors, working in radians provides more elegant and consistent formulas.

Can I calculate the radius if I know the chord length and arc length?

Yes, but it requires solving a transcendental equation that typically needs numerical methods. For practical purposes, it's easier to use this calculator to experiment with different radius values.

What's the maximum error when approximating arc with chord?

For a 90° angle, the chord is about 10% shorter than the arc. The approximation gets worse as the angle increases, with maximum difference at 180° where the chord is about 36% shorter than the semicircular arc.

Arc Length Calculator - Chord Sector Results

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Arc Length Calculator - Chord Sector Results

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