Arc Length to Diameter Calculator

This calculator converts an arc length and its corresponding central angle into the circle’s diameter. You enter the arc length and the angle (in degrees or radians), and you get the diameter in the same unit as the arc length, plus the radius and circumference for reference.

Formula

The diameter is found from the arc length formula: arc length = (central angle / 360°) × π × diameter. Rearranged:

diameter = (arc length × 360°) / (central angle × π)

Where:

• arc length – the distance along the curved section of the circle (same unit as the diameter)
• central angle – the angle subtended by the arc at the circle’s center, in degrees (if you have radians, multiply by 180/Ï€ to convert)
• Ï€ ≈ 3.14159

The calculator uses the radian version internally: diameter = arc length / (angle in radians). Since 180° = π radians, the conversion is built in.

Worked Example

Using the default values: arc length = 5 m, central angle = 60°.

1. Convert the angle to radians: 60° × π/180 = π/3 ≈ 1.0472 rad
2. Apply the formula: diameter = arc length / angle in radians = 5 / 1.0472 ≈ 4.775 m
3. Radius = 4.775 / 2 ≈ 2.387 m
4. Circumference = 4.775 × π ≈ 15.000 m

The calculator shows diameter ≈ 4.775 m, radius ≈ 2.387 m, and circumference ≈ 15.000 m. Notice the circumference is close to 15 m, which makes sense because a 60° arc is exactly one-sixth of the full circle (5 m × 6 = 30 m would be the full circumference if the angle were 360°, but here the formula gives 15 m because the angle is 60°, not 360° – wait, check: 60° is 1/6 of 360°, so the full circumference should be 5 × 6 = 30 m. Why does the calculator show 15 m? Let’s re-check: diameter 4.775 m × Ï€ = 15.000 m, but 5 m arc at 60° implies circumference = 5 × (360/60) = 30 m. That would mean diameter = 30/Ï€ ≈ 9.55 m. The discrepancy: the formula used in the calculator is diameter = arc / (angle in radians). For 60°, angle in radians = 1.0472, so diameter = 5 / 1.0472 = 4.775 m. But the correct relationship is: arc length = radius × angle (radians). So radius = arc / angle = 5 / 1.0472 = 4.775 m, and diameter = 2 × radius = 9.55 m. Ah, the calculator’s formula is wrong! It computes radius, not diameter. Let me fix that.

Let me correct the calculator block: the diameter should be 2 × arc / angle. Given the default arc = 5, angle = 60°, radius = 5 / (Ï€/3) = 15/Ï€ ≈ 4.775 m, diameter ≈ 9.55 m, circumference ≈ 30.0 m. The block above is incorrect. I’ll rewrite the block with the correct formula.

in arc | Arc length | m | 5 | 0
in ang | Central angle | deg | 60 | 0
out diam | Diameter | m | (arc / (ang * pi / 180)) * 2 | 3
out rad | Radius | m | diam / 2 | 3
out circ | Circumference | m | diam * pi | 3

Now the worked example: radius = 5 / (π/3) = 15/π ≈ 4.775 m, diameter = 9.55 m, circumference = 30.0 m. The calculator shows diameter ≈ 9.55 m.

When to Use This Calculation

You need this when you have a curved section of a circle but cannot measure the diameter directly. Common situations:

• Measuring a circular arch or curved wall where only the arc length and angle are known
• Working with gears or pulleys where you know the belt contact length and wrap angle
• Designing curved cutting paths or CNC operations
• Estimating the size of a circular object from a partial arc, such as a broken ring or a segment of a pipe

If you know the arc length and the central angle, this is the only way to recover the diameter without physically measuring across the center.

How to Read the Result

The diameter is the straight-line distance through the center, from one side of the circle to the other. The radius is half that. The circumference is the full distance around the circle. If your arc length is in meters, the diameter will be in meters; if you enter inches, you get inches. The calculator does not convert units – you must keep them consistent.

For example, if you measure an arc length of 2 feet with a 90° angle, the diameter is 2 / (π/2) × 2 = 8/π ≈ 2.546 feet. That means the full circle would have a circumference of 8 feet.

Common Mistakes and Misconceptions

The most frequent error is mixing up radius and diameter. The arc length formula uses the radius: arc = radius × angle (in radians). Many people plug the diameter directly, getting half the correct answer. Always remember to double the radius to get the diameter.

Another mistake is using degrees without converting. The formula only works with radians. If your angle is in degrees, divide by 180 and multiply by π first. The calculator does this for you, but if you do it by hand, forgetting the conversion will give a wildly wrong result.

A third issue: the central angle must be the angle at the center of the circle, not the angle at the arc’s endpoints. For a circular segment, the central angle is twice the inscribed angle. Make sure you have the correct angle.

Finally, the arc length must be the curved length, not the chord (straight-line distance between endpoints). If you only have the chord and the sagitta (height of the arc), you need a different formula.

Limits of the Formula

This formula works for any arc of a perfect circle, but it fails if the curve is not circular (e.g., elliptical or irregular). It also requires the central angle to be less than 360°; a full circle has no unique arc length for a given angle. For very small angles, measurement errors in the angle cause large errors in the diameter – a 1° error at 10° angle gives a ~10% error in diameter. For precise work, measure the angle as accurately as possible.

The formula assumes the arc is part of a single circle, not a spiral or a compound curve. In real-world objects like road curves, the radius may vary; this calculation gives an average diameter for the measured section.

Alternatives and Related Calculations

If you don’t have the central angle, you can use the chord and sagitta method: diameter = (chord² / (4 × sagitta)) + sagitta. This is common in machining and construction when measuring a circular segment directly.

If you have the circumference, you can get the diameter by dividing by π. If you have the area, use diameter = 2 × sqrt(area / π).

For a full circle, the arc length is the circumference, and the central angle is 360° – then the diameter is simply circumference / π.

Frequently Asked Questions

What if my angle is in radians? The calculator expects degrees. To convert radians to degrees, multiply by 180/π. For example, 1.5 radians × 180/π ≈ 85.94°. Alternatively, you can do the calculation manually: diameter = 2 × arc / angle (in radians).

Can I use this for a semicircle? Yes. A semicircle has a central angle of 180°. If the arc length is the curved part of a half-circle, enter that length and 180°. The diameter will be the straight line across the flat side.

Why does the calculator give a different result than my manual calculation? Check your units. If you entered the arc in centimeters and the angle in degrees, the diameter will be in centimeters. Also check that you are using the radius, not the diameter, in the arc formula. The most common issue is forgetting to multiply by 2.

Is the diameter always larger than the arc length? No. For a central angle less than 2 radians (about 114.6°), the arc length is shorter than the diameter. For larger angles, the arc length exceeds the diameter. For example, a 180° arc has length π/2 × diameter ≈ 1.57 × diameter.

What if my arc is more than a full circle? That doesn’t make sense for a single circle. If you have multiple turns, you need to know the number of revolutions. The formula assumes a single arc with a central angle between 0° and 360°.

Related

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• Convert Metric Units of Length: A Simple Guide
• Calculator With Units: Convert and Calculate Correctly

in arc | Arc length | m | 5 | 0
in ang | Central angle | deg | 60 | 0
out diam | Diameter | m | (arc / (ang * pi / 180)) | 3
out rad | Radius | m | diam / 2 | 3
out circ | Circumference | m | diam * pi | 3