Arc Length to Radius Calculator: Formula & Steps

This calculator finds the radius of a circle when you know the arc length and the central angle (in degrees or radians). You enter the arc length and the angle, and it returns the radius in the same unit you used for the arc length. The formula is simply radius = arc length / angle (in radians).

Formula

The relationship between arc length (s), radius (r), and central angle (θ in radians) is:

s = r × θ

Rearranged to solve for the radius:

r = s / θ

• s = arc length (same unit as the radius, e.g., meters, feet, inches)
• r = radius (same unit as the arc length)
• θ = central angle in radians (not degrees)

If your angle is in degrees, convert it first: θ (radians) = θ (degrees) × π / 180.

Worked Example

Using the default values from the calculator: arc length = 10 m, central angle = 60°.

1. Convert the angle to radians: 60 × π / 180 = π/3 ≈ 1.0472 radians.
2. Apply the formula: r = 10 / 1.0472 ≈ 9.549 m.
3. The calculator rounds to 2 decimal places: 9.55 m.

So a 10-meter arc that subtends a 60° angle belongs to a circle with a radius of about 9.55 meters.

When to Use This Calculation

You need this whenever you have a curved section and you want to know the size of the circle it came from. Common situations include:

• Road or track design – finding the turning radius from a measured curve.
• Machining and fabrication – determining the radius of a bent pipe or sheet metal part from the arc length of the bend.
• Woodworking and carpentry – laying out arches or curved moldings.
• Geography and surveying – estimating the Earth’s radius from a measured arc on a map (though the Earth is not a perfect sphere).
• Astronomy – calculating the radius of a planet’s orbit from the arc it sweeps in a given time (if the angle is known).

In short, any time you have a piece of a circle and you need the full circle’s size, this is the tool.

How to Read the Result

The result is the radius in the same unit you entered for the arc length. If you entered arc length in meters, you get the radius in meters; if you entered feet, you get feet. The angle unit does not affect the output unit – it only affects the numeric conversion.

For example, an arc length of 5 feet with a 90° angle gives a radius of 5 / (π/2) ≈ 3.18 feet. If you instead entered 5 meters, you would get 3.18 meters. The unit always follows the arc length.

If the angle is very small (say 1°), the radius will be large relative to the arc length. If the angle is large (say 300°), the radius will be small. This makes sense: a gentle curve comes from a big circle; a tight curve comes from a small one.

Common Mistakes and How to Avoid Them

• Using degrees without converting – The formula only works with radians. If you plug in degrees directly, you will get a wrong answer (usually far too large). Always convert degrees to radians first.
• Mixing units – The arc length and the radius must be in the same unit. If your arc length is in inches and you want the radius in feet, convert one of them before calculating.
• Confusing arc length with chord length – The arc length is the distance along the curve, not the straight-line distance between the two endpoints (chord). Using the chord length will give a different, incorrect radius.
• Rounding too early – If you round the angle in radians to 2 decimal places before dividing, you may get a noticeably different result. Keep full precision until the final step.

Limitations of the Formula

The formula assumes a perfect circle – a constant radius. Real-world objects like roads or bent pipes may not be perfect arcs; they might be spirals or have varying curvature. In those cases, this calculation gives an average or approximate radius, not an exact one.

Also, the angle must be between 0 and 360 degrees (or 0 and 2π radians) for a single arc. If you have an angle larger than 360°, you are describing more than one full revolution, which is not a simple arc. The formula still works mathematically, but the interpretation changes.

For very small angles (less than a few degrees), the arc length is nearly equal to the chord length, but the formula remains accurate as long as you use the true arc length.

Alternative Methods

If you do not have the central angle, you can find the radius using other measurements:

• Chord length and height – If you know the straight-line distance between the arc’s endpoints (chord) and the maximum height of the arc above the chord, the radius is given by: r = (h/2) + (c² / (8h)), where h is the height and c is the chord length.
• Three points on the arc – If you know the coordinates of three points on the circle, you can find the circle’s center and radius using perpendicular bisectors.
• Circumference – If you know the full circumference, the radius is simply C / (2Ï€). This is the easiest but requires the whole circle, not just an arc.

These alternatives are useful when the central angle is hard to measure directly.

Frequently Asked Questions

Can I use this calculator for angles in radians?

Yes. The calculator has a field for the angle in degrees, but you can enter radians by converting: if your angle is 1 radian, enter 57.2958 degrees (since 1 rad = 180/Ï€ degrees). Alternatively, you can do the math manually: radius = arc length / angle in radians.

What if I only know the arc length and the chord length?

You cannot directly use this calculator because it needs the central angle. However, you can derive the angle using trigonometry: θ = 2 × arcsin(chord / (2r)), but that requires solving for r iteratively. Easier: use the chord-height formula mentioned above if you also have the height.

Why does the formula require radians and not degrees?

Radians are a natural unit where the arc length equals the radius times the angle. In degrees, the formula would be s = r × θ × (π/180), which is less elegant. Using radians keeps the relationship simple and is the standard in mathematics and physics.

Does the radius change if I use a different unit for the arc length?

No, the radius is always in the same unit as the arc length. If you switch from meters to feet, the numeric value changes (because 1 m ≈ 3.28 ft), but the physical circle is the same. The calculator simply outputs the radius in the unit you entered.

Can I find the diameter from the radius?

Yes, the diameter is simply twice the radius. After getting the radius from this calculator, multiply by 2 to get the diameter. For example, a radius of 9.55 m gives a diameter of 19.10 m.

What if my arc is more than half a circle?

The formula still works. For an arc length of 20 m with a 300° angle, the radius would be 20 / (300 × π/180) ≈ 3.82 m. The calculator handles angles up to 360° without issue.

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in arc | Arc Length | m | 10 | 0
in ang | Central Angle | deg | 60 | 0
out r | Radius | m | arc / (ang * pi / 180) | 2
out a | Angle in Radians | rad | ang * pi / 180 | 3