Arc Segment Length Calculator

This calculator finds the length of an arc segment of a circle. You enter the radius and the central angle (in degrees or radians), and it returns the arc length in the same unit you used for the radius.

Formula

The arc length s is the distance along the curved line that makes up the arc. For a circle of radius r, the full circumference is 2πr. The arc is a fraction of that circumference, equal to the central angle divided by the full angle of 360° (or 2π radians).

In degrees: s = (π × r × θ) / 180

In radians: s = r × θ

Where:

• s = arc length (same unit as radius)
• r = radius of the circle (any unit: m, cm, ft, in, etc.)
• θ = central angle in degrees (for the first formula) or radians (for the second)

Worked Example

Using the default values from the calculator: radius = 5 m, angle = 60°.

1. Convert the angle to radians: 60° × (π / 180) = π/3 ≈ 1.0472 radians.
2. Apply the radian formula: s = r × θ = 5 × 1.0472 ≈ 5.236 m.
3. Or apply the degree formula directly: s = (π × 5 × 60) / 180 = (300π) / 180 = (5π)/3 ≈ 5.236 m.

The calculator returns 5.24 m (rounded to two decimals), matching the result above.

When to Use Arc Length

Arc length appears in many practical situations:

• Laying out a curved road or railway track – the arc length is the actual distance along the curve.
• Designing gears or pulleys – the length of belt that contacts a pulley is an arc length.
• Creating circular patterns in woodworking or metalworking – knowing the arc length helps cut or bend material accurately.
• Converting between linear and angular motion – for example, a point on a rotating wheel travels an arc length equal to radius × angle (in radians).
• Land surveying – boundary lines that follow a circular curve are measured as arc lengths.

If you have the diameter instead of the radius, simply divide by 2. If you have the angle in radians, use the simpler formula s = r × θ.

How to Read the Result

The result is a linear distance, so it carries the same unit as the radius you entered. If you enter the radius in meters, the arc length is in meters; if you enter feet, you get feet. This is straightforward, but it’s a common point of confusion when the angle is in degrees – the degree formula already includes the conversion factor Ï€/180, so you do not need to convert the angle yourself.

For example, a 90° arc on a circle of radius 10 cm has length (π × 10 × 90)/180 = 5π ≈ 15.71 cm. That is one-quarter of the circumference (2π × 10 = 62.83 cm; 62.83/4 = 15.71 cm).

Common Mistakes and Misconceptions

• Mixing units: The radius and the arc length must use the same unit. If you measure the radius in inches but want the result in feet, convert first.
• Using diameter instead of radius: The formula uses radius. If you have diameter, halve it before entering.
• Confusing arc length with chord length: The chord is the straight line connecting the two endpoints of the arc. The arc length is always longer than the chord (unless the angle is 0°). For a 60° arc on a 5 m radius, the chord is 2 × 5 × sin(30°) = 5 m, while the arc is 5.24 m.
• Forgetting to convert degrees to radians when using the radian formula: If you use s = r × θ, the angle must be in radians, not degrees. The calculator handles this for you.
• Assuming the angle is the full circle: If the angle is 360°, the arc length equals the full circumference. If the angle is 180°, it’s half the circumference, and so on.

Limits and Accuracy

The formula assumes a perfect circle and a constant radius. Real-world objects may deviate slightly, and if the radius varies along the arc, the simple formula no longer applies – you would need calculus or numerical approximation. Also, the angle must be measured in the same plane as the circle; if you’re working with a 3D curve, this is not the right tool.

Rounding: The calculator rounds to two decimal places for the arc length. If you need more precision for precise manufacturing, increase the decimal places or use the exact expression (π × r × θ)/180 without rounding until the final step.

Frequently Asked Questions

What is the difference between arc length and chord length?

Arc length is the distance along the curved line of the circle between two points. Chord length is the straight-line distance between those same two points. For any arc greater than 0°, the arc length is larger than the chord. For a semicircle (180°), the chord is the diameter, and the arc length is half the circumference.

Can I use this calculator for a sector’s perimeter?

Yes, but you need to add the two radii. The perimeter of a sector is arc length + 2 × radius. For example, with radius 5 m and angle 60°, the arc length is 5.24 m, so the sector perimeter is 5.24 + 10 = 15.24 m.

What if my angle is in radians?

You can still use the calculator by converting the angle to degrees first (multiply by 180/Ï€), or you can use the formula s = r × θ directly. The calculator’s default is degrees, but you can enter a radian value after converting it manually.

Does the arc length depend on the angle unit?

No, the physical arc length is the same regardless of whether you express the angle in degrees or radians. The formula adjusts automatically. Using degrees requires the factor π/180; using radians gives the simpler s = r × θ because radians are already a ratio of arc length to radius.

Why is the arc length formula different for radians?

By definition, an angle of 1 radian subtends an arc equal in length to the radius. So the arc length is simply radius times the number of radians. Degrees are an arbitrary unit (360 per full circle), so they require a conversion factor.

What if I only know the chord length and the radius?

You can find the central angle using trigonometry. For chord length c and radius r, the angle θ (in radians) is 2 × arcsin(c / (2r)). Then use s = r × θ. This is a common situation in construction and design.

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in r | Radius | m | 5 | 0
in a | Central angle | deg | 60 | 0
out s | Arc length | m | (pi * r * a) / 180 | 2
out rad | Angle in radians | rad | a * (pi / 180) | 4