Circle Arc Length Calculator: Formula & Examples

This calculator works out the length of a circular arc from the circle’s radius and the angle that the arc subtends at the center. You enter the radius and the angle (in degrees or radians), and you get the arc length in the same unit as the radius, plus the sector area and chord length. It is a direct application of the formula s = rθ, with the angle converted to radians when needed.

Formula

The arc length s of a circle is found using the formula:

s = r × θ (when θ is in radians)

If the angle is given in degrees, convert it first: θ (radians) = θ (degrees) × π / 180. Then the formula becomes:

s = (π × r × a) / 180

where:

• s = arc length (same unit as the radius, e.g., meters, feet, inches)
• r = radius of the circle (any unit of length)
• a = central angle in degrees (the angle between the two radii that touch the arc’s endpoints)
• Ï€ ≈ 3.14159

The sector area (the pie-slice shape) is given by A = (Ï€ × r² × a) / 360 and the chord length (the straight distance between the arc’s endpoints) is c = 2r × sin(a/2).

Worked Example

Using the default values from the calculator: radius r = 5 m and central angle a = 60°.

1. Convert the angle to radians: 60° × π/180 = π/3 ≈ 1.0472 radians.
2. Apply the formula s = r × θ = 5 × 1.0472 ≈ 5.236 m.
3. Using the degree-based formula directly: s = (π × 5 × 60) / 180 = (300π) / 180 = 5π/3 ≈ 5.236 m.
4. Rounded to two decimals, the arc length is 5.24 m.

The calculator also gives the sector area: (π × 25 × 60) / 360 = 1500π/360 ≈ 13.09 m², and the chord length: 2 × 5 × sin(30°) = 10 × 0.5 = 5.00 m.

When to Use Arc Length

Arc length is used whenever you need the distance along a curved part of a circle, not the straight line. Common situations include:

• Finding the length of a curved road or path that follows a circular arc.
• Calculating the length of material needed to make a circular trim, molding, or bent tube.
• Determining the distance traveled by a point on a rotating wheel (for one full revolution, the arc length equals the circumference).
• Working with gear teeth, pulleys, or conveyor belts that wrap around a circular part.
• Solving geometry problems in school, especially those involving sectors or segments of circles.

If the angle is a full circle (360°), the arc length equals the circumference: 2πr. For a semicircle (180°), it is πr.

How to Read the Result

The arc length is always in the same linear unit as the radius. If you enter the radius in meters, you get the arc length in meters; if you enter inches, you get inches. The sector area is in square units (e.g., m², ft²), and the chord length is also in the same linear unit as the radius.

For example, a radius of 10 cm with a 90° angle gives an arc length of (π × 10 × 90)/180 = 15.708 cm. That is one-quarter of the circumference (2π × 10 = 62.832 cm, divided by 4).

If the result seems too large or too small, check that the angle is in degrees, not radians. A common mistake is entering radians as if they were degrees, which would give an arc length that is off by a factor of 180/π ≈ 57.3.

Common Mistakes and Pitfalls

• Mixing up degrees and radians: The formula s = rθ only works with radians. If you use degrees without converting, your answer will be wrong. Always convert degrees to radians by multiplying by Ï€/180.
• Using diameter instead of radius: The formula uses the radius. If you have the diameter, divide it by 2 first.
• Forgetting to use the same units: If the radius is in feet and the arc length is needed in inches, convert the radius to inches before calculating.
• Rounding too early: Rounding Ï€ to 3.14 or rounding intermediate steps can cause noticeable errors. Keep at least 4-5 decimal places in intermediate steps.
• Confusing arc length with chord length: The arc is the curved part; the chord is the straight line between the endpoints. They are equal only when the angle is 0°.

Limits of the Formula

The formula s = rθ is exact for a perfect circle. It assumes the arc is part of a circle with constant radius. Real-world objects may not be perfect circles, so the result is an approximation for non-circular curves. Also, the angle must be positive and less than or equal to 360° (or 2π radians) for a single arc; angles larger than that represent more than one full revolution, and the arc length would be the sum of multiple circumferences plus the remainder.

For very large angles (e.g., 720°), the formula still works mathematically, but it represents two full turns. In practice, most arc length problems use angles between 0° and 360°.

Frequently Asked Questions

What is the formula for arc length?

The arc length s is the radius r multiplied by the central angle θ in radians: s = rθ. If the angle is in degrees, use s = (π × r × a) / 180, where a is the angle in degrees.

How do I find the arc length without the radius?

If you don’t have the radius, you can find it from the circumference (C = 2Ï€r) or from the chord length and the angle. If you know the chord length c and the central angle a (in degrees), the radius is r = c / (2 × sin(a/2)). Then use the arc length formula.

What is the difference between arc length and chord length?

Arc length is the distance along the curved line of the circle. Chord length is the straight-line distance between the two endpoints of the arc. The chord is always shorter than the arc, except when the angle is 0° (both are zero). For a 180° arc (semicircle), the chord is the diameter (2r) and the arc length is πr.

Can I use this calculator for a full circle?

Yes. For a full circle, use a central angle of 360°. The arc length will be 2πr, which is the circumference. For example, with radius 5 m, the arc length is 31.42 m.

What units should I use?

Any units for the radius work, as long as you are consistent. The arc length will be in the same unit as the radius. The angle can be in degrees or radians, but the calculator on this page expects degrees. If you have radians, multiply by 180/Ï€ to convert to degrees before entering.

Why is the arc length formula s = rθ?

This comes from the definition of a radian: one radian is the angle that subtends an arc equal in length to the radius. So the total arc length is the number of radians times the radius. Since a full circle is 2Ï€ radians, the arc length for a full circle is 2Ï€r, which matches the circumference formula.

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in r | Radius | m | 5 | 0
in a | Central angle | degrees | 60 | 0
out s | Arc length | m | (pi * r * a) / 180 | 2
out area | Sector area | m^2 | (pi * r^2 * a) / 360 | 2
out chord | Chord length | m | 2 * r * sind(a / 2) | 2