Arc Length to Angle: Formula & Calculator

This calculator converts an arc length and radius into the central angle it subtends, giving the result in both degrees and radians. You enter the arc length and the circle’s radius, and you get back the angle in degrees (the primary answer) plus the angle in radians. The relationship is simply angle = arc length / radius, which is the definition of a radian.

Formula

The core formula connecting arc length, radius, and angle is:

θ (radians) = s / r

To convert to degrees, multiply by 180/π:

θ (degrees) = (s / r) × (180 / π)

Where:

• s = arc length (same unit as radius, e.g., meters, feet)
• r = radius of the circle (same unit as arc length)
• θ = central angle (in radians for the first formula, degrees for the second)

This works because a radian is defined as the angle subtended by an arc whose length equals the radius. So if the arc is twice as long as the radius, the angle is 2 radians.

Worked Example

Using the calculator defaults: arc length s = 5 m, radius r = 10 m.

1. First, find the angle in radians: θ = s / r = 5 / 10 = 0.5 radians.
2. Convert to degrees: θ = 0.5 × (180 / π) ≈ 0.5 × 57.2958 ≈ 28.65°.

The calculator shows 28.65° and 0.5000 radians, matching the manual calculation.

Why Radians Matter

Radians are the natural unit for angles in mathematics and physics. Unlike degrees, which are an arbitrary division of a circle into 360 parts, radians are dimensionless: they are the ratio of two lengths (arc length / radius). This makes them essential in calculus, where derivatives of trigonometric functions are simple only when angles are in radians. For example, the derivative of sin(x) is cos(x) only if x is in radians. In engineering and physics, angular velocity (radians per second) and angular acceleration (radians per second squared) are standard because they simplify rotational equations.

When to Use This Conversion

You might need to convert arc length to angle in many practical situations:

• Gear and pulley design: Finding the angle of rotation from the linear distance a belt or chain moves.
• Navigation and surveying: Converting a measured distance along a curved path (e.g., a road or a meridian) into an angular change.
• Physics problems: Calculating the angular displacement of a wheel or pendulum from the distance traveled along its rim.
• Geometry homework: Finding the central angle of a sector given its arc length and radius.
• Machining and CNC: Determining the angular rotation needed to cut an arc of a given length on a workpiece.

In all these cases, the formula is the same; only the units of length might differ (meters, feet, inches).

Common Mistakes and How to Avoid Them

One common error is mixing up units. If the arc length is in centimeters and the radius is in meters, you must convert both to the same unit before dividing. For example, if s = 50 cm and r = 0.5 m, convert s to 0.5 m (or r to 50 cm) before dividing.

Another frequent mistake is forgetting to convert from radians to degrees when the problem expects degrees. Many calculators have a mode setting (DEG vs RAD) that can cause errors if not set correctly. Always check what units your final answer should be in.

A third issue is using the diameter instead of the radius. The formula requires the radius, not the diameter. If you only have the diameter, divide it by 2 first.

Finally, remember that the angle must be in radians for the formula θ = s / r to work directly. If you have the angle in degrees, you need to convert it to radians first (multiply by π/180) before using it to find arc length.

Limitations and Special Cases

The formula θ = s / r assumes a circular arc. If the path is not circular (e.g., an ellipse or a spiral), this simple relationship does not hold. In those cases, you would need calculus or numerical integration.

Also, the angle can be greater than 2π radians (360°) if the arc wraps around the circle more than once. For example, an arc length of 3πr corresponds to 3π radians (540°). The formula still works, but some calculators might return an angle in the range 0 to 2π; you may need to add multiples of 2π to get the total angle.

For very small angles, the arc length is nearly equal to the chord length (the straight line between the two endpoints). But the formula above is exact for any arc length, regardless of size.

Frequently Asked Questions

How do I find the angle from arc length and radius?
Divide the arc length by the radius. This gives the angle in radians. To get degrees, multiply by 180/π. For example, an arc of 3 meters on a circle of radius 2 meters gives 1.5 radians, which is about 85.94°.

What is the relationship between arc length and angle?
The arc length is directly proportional to the angle (in radians) and the radius: s = r × θ (θ in radians). Doubling the angle doubles the arc length, and doubling the radius also doubles the arc length.

Can I use this formula for a full circle?
Yes. For a full circle, the arc length is the circumference (2πr). The formula gives θ = 2πr / r = 2π radians, which equals 360°. That is consistent.

What is the difference between arc length and chord length?
Arc length is the distance along the curved path of the circle. Chord length is the straight-line distance between the two endpoints of the arc. For a given angle, chord length = 2r × sin(θ/2), which is always less than or equal to the arc length.

How do I convert radians to degrees?
Multiply the radians by 180/π. For example, 1 radian ≈ 57.2958°. Conversely, to convert degrees to radians, multiply by π/180.

What if I only know the arc length and the angle?
You can rearrange the formula to find the radius: r = s / θ (θ in radians). For example, if s = 4 meters and θ = 2 radians, then r = 2 meters.

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in s | Arc Length | m | 5 | 0
in r | Radius | m | 10 | 0
out deg | Angle (degrees) | ° | (s / r) * (180 / pi) | 2
out rad | Angle (radians) | rad | s / r | 4