Arc Length Calculator: Formula and Examples

An arc length calculator finds the distance along a circular curve from its radius and central angle. Enter the radius and angle, choose the angle unit, and it returns the arc length in the same units as the radius.

Formula

For an angle measured in degrees, arc length is s = r × θ × π / 180. Here, s is the arc length, r is the circle’s radius, and θ is the central angle in degrees. The equivalent formula when the angle is in radians is s = rθ. In either form, the radius and arc length use the same unit, such as inches, feet, millimeters, or meters.

The calculator accepts degrees and applies the degree-to-radian conversion, π / 180, within the formula. Its second result, the chord length, is the straight-line distance between the arc’s endpoints: c = 2r sin(θ / 2). The chord is not the curved arc length.

Worked Example

Using the calculator’s default inputs, the radius is 5 units and the central angle is 60 degrees.

1. Convert the angle to radians: 60 × π / 180 = π / 3 radians.
2. Calculate the arc length: s = 5 × π / 3 = 5π / 3 ≈ 5.23599 units.
3. Round to two decimal places, as the calculator does: arc length = 5.24 units.
4. For comparison, the chord is 2 × 5 × sin(30°) = 5.00 units.

The answer is shorter than the full circumference, 2πr ≈ 31.42 units, because 60 degrees is one-sixth of a full 360-degree turn.

What Arc Length Measures

A circle’s circumference measures the entire boundary. An arc is only a portion of that boundary, and its length measures the distance along the curve between two points. The central angle is the angle formed at the circle’s center by lines drawn to those endpoints.

Arc length grows in direct proportion to both radius and angle. Double the radius while keeping the angle fixed, and the arc doubles. Double the angle while keeping the radius fixed, and the arc doubles as well, provided you are considering the corresponding sweep around the circle.

For an angle expressed as a fraction of a full turn, another useful form is s = (θ / 360°) × 2πr. For example, a 90-degree arc is one quarter of the circumference. This is the same relationship as the degree formula, written to emphasize the fraction of the circle.

How to Use the Calculator

• Measure or identify the circle’s radius: the distance from its center to its edge.
• Enter the central angle that sweeps out the arc, in degrees.
• Read the arc length in the same length unit used for the radius. The chord result is a separate, straight-line measurement.
• Keep more digits for intermediate calculations if you need a precise final result; round the displayed value to suit the task.

If you know the diameter rather than the radius, divide it by two before entering it. For example, a circle with a 10-centimeter diameter has a radius of 5 centimeters. The calculator’s length unit is a label rather than a conversion: entering a radius in centimeters gives an answer in centimeters, while entering feet gives an answer in feet.

Degrees, Radians, and Units

Degrees divide a full rotation into 360 parts. Radians measure angles using the relationship between arc length and radius: one radian is the angle that subtends an arc equal in length to the radius. A full circle is 2π radians, and 180 degrees equals π radians.

Many calculators and mathematical formulas use radians, so check the angle mode when comparing results. If your known angle is already in radians, use s = rθ directly; do not multiply by π / 180 again. If the angle is in degrees, convert it once, either by multiplying by π / 180 or by using the degree version of the formula.

Metric and US customary units work in exactly the same way. A radius of 2 meters produces an arc length in meters; a radius of 2 feet produces one in feet. To convert a result, use the appropriate length conversion after calculating, or convert the radius first. Never combine a radius in one unit with a length result interpreted in another without converting.

Common Mistakes and Limits

• Using diameter as radius: This makes the calculated arc twice as long as it should be. Divide the diameter by two first.
• Confusing arc and chord: The arc follows the circle; the chord cuts straight across. For a nonzero minor arc, its length is greater than the chord joining its endpoints.
• Mixing angle units: The degree formula and radian formula require different forms. A missed or repeated conversion changes the answer.
• Using a sector area formula: Arc length is a distance, not an area. Sector area has square units and uses a different formula.
• Rounding too early: Keep π and intermediate values unrounded where possible, then round the final result.

The calculation assumes a perfect circle and a known radius and central angle. It does not measure a physical curve, compensate for measurement uncertainty, or determine the angle from an image. An angle greater than 180 degrees can describe a major arc; the formula still works if the intended sweep is clear. At 360 degrees, it gives the full circumference.

Frequently Asked Questions

How do I calculate arc length by hand?
For degrees, multiply the radius by the angle and by π / 180. For radians, multiply the radius directly by the angle. Keep the radius and answer in the same length units.

Can arc length be longer than the diameter?
Yes. Arc length measures distance along a curve, so it can exceed the circle’s diameter. A semicircle’s arc, for example, has length πr, while the diameter is 2r.

What is the difference between arc length and chord length?
Arc length follows the circle’s curved edge. Chord length is the straight segment connecting the arc’s endpoints. For a minor arc, the chord is shorter than the arc.

What if I know the arc length and want the angle?
Rearrange the radian formula to θ = s / r. The result is in radians. Convert it to degrees by multiplying by 180 / π.

Does a larger radius always make the arc longer?
For the same central angle, yes: arc length is directly proportional to radius. If the angle changes too, compare both values using s = rθ in radians.

Related

• Absolute Minimum and Maximum Calculator Explained
• Calculate the Golden Ratio
• Calculate the Hypotenuse of a Triangle

in r | Radius | units | 5 | 0
in a | Central angle | degrees | 60 | 0
out s | Arc length | units | r * a * pi / 180 | 2
out c | Chord length | units | 2 * r * sind(a / 2) | 2