Arc Length Calculator for Radians and Circle Arcs

An arc length calculator for radians finds the distance along a circular arc when you enter the circle’s radius and its central angle in radians. It returns the arc length in the same unit as the radius, along with the angle converted to degrees for reference.

Formula

The arc length formula for an angle measured in radians is s = rθ. Here, s is the length of the arc, r is the circle’s radius, and θ is the central angle in radians. The radius and arc length use the same length unit; radians are dimensionless.

A central angle is the angle formed at the circle’s center by two radii that meet the endpoints of the arc. The formula works for minor arcs and major arcs: use the angle that corresponds to the arc you want. For a full circle, θ = 2π radians, so s = 2πr, the circumference.

Worked Example

Using the calculator’s defaults, the radius is 5 units and the central angle is 1.2 radians.

1. Apply the radian formula: s = rθ.
2. Substitute the values: s = 5 × 1.2 = 6 units.
3. Convert the angle to degrees: 1.2 × 180 / π ≈ 68.75°.

The arc length is 6.00 units, and the central angle is about 68.75°. Because the calculator’s radius unit is “units,” the result is also in units. If you enter a radius in centimeters, the result is centimeters; if you enter feet, it is feet.

Why Radians Make the Formula Simple

A radian measures an angle by comparing the length of an arc with the radius. One radian is the angle that subtends an arc equal in length to the radius. In symbols, θ = s/r; rearranging gives s = rθ directly.

Degrees describe the same angles using a different scale: a complete turn is 360°, while it is 2π radians. That is why degree-based calculations need a conversion factor. If an angle is in degrees, convert it first:

• Radians = degrees × π / 180
• Degrees = radians × 180 / π

For example, a 90° angle is π/2 radians. With a radius of 4 meters, its arc length is 4 × π/2 = 2π meters, approximately 6.28 meters.

How to Use the Calculator

• Enter the circle’s radius, not its diameter. The radius is the distance from the center to the circumference.
• Enter the central angle in radians. If your angle is in degrees, convert it using the formula above before entering it.
• Read the first result as the arc’s length. It uses the same unit as the radius.
• Use the degree conversion as a check or for comparison with a measurement given in degrees.

The example inputs are just defaults. Replace them with the measurements for your circle. A radius must be greater than zero for an ordinary circle, and the angle should represent the portion of the circle whose arc you need.

Choosing the Correct Angle and Units

The angle must be the central angle, not an angle measured at a point on the circumference. If a problem gives an inscribed angle—the angle whose vertex lies on the circle—the corresponding central angle is twice as large when both angles intercept the same arc. Check the diagram or wording before calculating.

Also decide whether the requested arc is the shorter or longer path between two points. The minor arc usually has a central angle below π radians (180°); the major arc uses the remaining angle around the circle. For example, if the minor angle is 1 radian, the major arc’s angle is 2π − 1 radians.

Keep the radius and final answer in compatible units. The calculation does not convert units: a radius of 2 feet and an angle of 1 radian produce an arc length of 2 feet. If you need metric output, convert the radius first; 1 foot equals 0.3048 meter. Similarly, convert a radius in inches to centimeters or meters before calculating if that is the desired output unit.

Common Mistakes and Limits

• Entering degrees as radians: A value of 90 means 90 radians if entered in the angle field, not 90°. Convert 90° to π/2 radians first.
• Using the diameter: If only the diameter is known, divide it by 2 to get the radius.
• Mixing length units: The answer follows the radius unit; the formula will not detect or fix an accidental unit mismatch.
• Rounding too early: Keep π or the full converted angle in intermediate calculations, then round the final result to the precision justified by the measurements.
• Choosing the wrong arc: A pair of endpoints can define a minor arc and a major arc. Use the angle for the intended path.

The formula describes a circular arc with a constant radius. It does not directly calculate the length of an irregular curve, an ellipse’s arc, or a path whose radius changes. Those shapes require different methods, often involving geometry or numerical approximation. Measurement uncertainty in the radius and angle also carries through to the answer, so extra decimal places do not make uncertain inputs more precise.

Frequently Asked Questions

How do you calculate arc length with radians?
Multiply the circle’s radius by the central angle in radians: s = rθ. The result is in the same length unit as the radius.

Can I use degrees in s = rθ?
Not directly. Convert degrees to radians by multiplying by π/180, then multiply the converted angle by the radius.

What is the arc length of a full circle in radians?
A full turn is 2π radians, so its arc length is r × 2π, which is the circumference 2πr.

Is arc length the same as circumference?
No. Arc length is the distance along one portion of a circle. Circumference is the distance around the entire circle and is the arc length for a 2π-radian angle.

Does the radius need to be in meters?
No. Use any consistent length unit, such as millimeters, centimeters, inches, feet, or meters. The arc length will be in that same unit.

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in r | Radius | units | 5 | 0
in theta | Central angle | radians | 1.2 | 0
out s | Arc length | units | r * theta | 2
out degrees | Angle | degrees | theta * 180 / pi | 2