Arc Circumference Calculator: Find the Arc Length

An arc circumference calculator works out the length of a circle’s arc from its radius and central angle. Enter the radius and angle in degrees; the calculator returns arc length in the same distance unit as the radius, along with the full circumference for comparison.

Formula

For a central angle measured in degrees, arc length is s = (π × r × θ) / 180. Here, s is the arc length, r is the circle’s radius, and θ (theta) is the central angle in degrees. The radius and arc length use the same distance unit, such as inches, feet, centimeters, or meters.

The full circumference is C = 2πr. A circle’s arc is the portion of its circumference corresponding to the angle, so the same relationship can be written as s = (θ / 360) × C. The calculator uses the radius-and-angle form and shows the circumference as a helpful reference.

Worked Example

Using the calculator’s default values, the radius is 5 units and the central angle is 90 degrees.

1. Substitute the values into the arc-length formula: s = (π × 5 × 90) / 180.
2. Since 90 / 180 = 0.5, s = 2.5π.
3. Calculate and round to two decimal places: s ≈ 7.85 units.
4. For comparison, C = 2π × 5 = 10π ≈ 31.42 units.

The arc is one quarter of the circumference because 90 degrees is one quarter of 360 degrees. One quarter of 31.42 is approximately 7.85, which checks the result.

What Arc Length and Circumference Mean

The circumference is the distance all the way around a circle. An arc is a selected part of that circular boundary, measured along the curve rather than across the interior. The central angle is formed by two radii that meet at the circle’s center; it determines what fraction of the full circle the arc covers.

For example, a 60-degree central angle selects 60/360, or one-sixth, of a circle. A 180-degree angle produces a semicircle, while 360 degrees traces the entire circumference. The formula also works for angles greater than 180 degrees when you mean the longer arc, up to a full 360-degree revolution.

When to Use the Calculator

Use an arc circumference calculator when you know a circle’s radius and the angle at its center but need the curved distance between the two points where the radii meet the circle. Applications include geometry exercises, circular layouts, portions of wheels or tracks, and measuring a sector’s curved edge.

• Choose the radius, not the diameter. If you know the diameter, divide it by 2 first.
• Measure the angle at the center of the circle, not at a point on its edge.
• Keep the radius and desired length in matching units. A radius in centimeters gives an arc length in centimeters.
• Decide whether you want the minor arc (the shorter path) or major arc (the longer path); use the corresponding central angle.

Degrees, Radians, and Unit Choices

This calculator accepts degrees. If an angle is supplied in radians, do not enter it as though it were degrees. The radian version of the formula is s = rθ, where θ is in radians. To convert degrees to radians, multiply the degree measure by π/180; to convert radians to degrees, multiply by 180/π.

Distance units do not change the geometry. A radius of 5 inches and an angle of 90 degrees give an arc of about 7.85 inches; a radius of 5 meters with the same angle gives about 7.85 meters. For US customary measurements, use inches, feet, or yards consistently. For metric measurements, use millimeters, centimeters, or meters consistently. Convert the final answer separately if a different output unit is needed.

Common Errors and Limits

A frequent mistake is using the diameter in place of the radius. Because the diameter is twice the radius, that error makes the calculated arc twice as long as it should be. Another is mixing units—for example, entering a radius in feet while expecting a result in inches without converting.

Rounding π or intermediate values too early can also slightly change the result. Keep the calculator’s precision through the calculation and round the final length to a sensible number of decimal places for the measurement. The displayed result is a mathematical value; the precision of a real measurement cannot exceed the accuracy of the radius and angle you entered.

The formula assumes a circular arc with a known, constant radius. It does not find an irregular curve’s length, and it cannot determine the radius or angle from nothing. If the angle is measured from a different reference or the endpoints are not on the same circle, establish the correct central angle and radius first.

Frequently Asked Questions

How do I calculate arc length from circumference?
Multiply the full circumference by the fraction of the circle represented by the central angle: s = C × θ/360 when θ is in degrees. For example, a 90-degree arc is one quarter of its circle’s circumference.

Can I calculate arc length with diameter instead of radius?
Yes. Since radius is half the diameter, substitute r = d/2 into the formula: s = π × d × θ/360 for an angle in degrees.

Does the angle need to be in degrees?
This calculator expects degrees. If you have radians, convert the angle to degrees first or use s = rθ with the angle in radians.

What is the difference between arc length and chord length?
Arc length follows the curved edge of the circle. Chord length is the straight-line distance between the arc’s endpoints. They are not the same except in the limiting case of a very small arc.

How do I find the length of the major arc?
Use the central angle for the longer arc. If you know the minor angle θ, the major angle is 360° − θ; enter that value with the same radius.

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in r | Radius | units | 5 | 0
in a | Central angle | degrees | 90 | 0
out s | Arc length | units | pi * r * a / 180 | 2
out c | Full circumference | units | 2 * pi * r | 2