Curve Calculator: Radius & Arc Length

This calculator works out the radius of a circle from either the arc length and central angle, or the chord length and sagitta (height), and then computes the arc length, chord length, and sagitta for any circle. You enter two known values, and the calculator returns the remaining circle measurements, including the radius and arc length.

Formula

The core formula for arc length is s = r × θ, where:

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

To use degrees, convert: θ (radians) = θ (degrees) × π / 180. So the radius from arc length and angle is:

r = (s × 180) / (θ × π)

For the chord length (the straight line between two points on the circle) and sagitta (the height of the arc from the chord midpoint to the circle), use:

• Chord length: c = 2 × r × sin(θ/2) (θ in radians)
• Sagitta: h = r − r × cos(θ/2) (θ in radians)

All lengths share the same unit; the angle is always in degrees in the calculator, but the formulas above use radians for clarity.

Worked Example

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

1. Convert angle to radians: 60 × π / 180 = π/3 ≈ 1.0472 rad
2. Radius: r = 10 / (π/3) = 30/π ≈ 9.549 m
3. Chord length: c = 2 × 9.549 × sin(30°) = 19.098 × 0.5 = 9.549 m
4. Sagitta: h = 9.549 − 9.549 × cos(30°) = 9.549 − 9.549 × 0.8660 = 9.549 − 8.268 = 1.281 m

The calculator rounds the radius to 9.549 m, chord to 9.549 m, and sagitta to 1.281 m. Notice that when the angle is 60°, the chord equals the radius—a nice check.

When to Use This Calculator

You might need the radius and arc length for:

• Road and rail design – horizontal curves on highways and railways are defined by radius and arc length.
• Architecture and construction – arched windows, bridges, or curved walls require accurate radius and arc measurements.
• Manufacturing – cutting curved metal or wood pieces, or designing gears and pulleys.
• Surveying – measuring curved property boundaries or laying out circular features.
• Everyday geometry – finding the size of an arch or a circular segment in a craft project.

If you only know the chord length and the sagitta (height of the arc), you can find the radius with the formula: r = (c² / (8h)) + (h/2). Many users come with those two measurements, so this calculator also accepts them in the inputs (though the current version uses arc length and angle).

How to Read the Results

The first output, Radius, is the key value—it defines the circle. The arc length is what you entered (if you entered it), but the calculator also gives the chord and sagitta, which are useful for physical layout. For example, if you know the radius and angle, you can compute the arc length directly. If you are building an arch with a 60° opening and a 10 m arc, the radius is 9.549 m, meaning the center of the circle is 9.549 m from any point on the arc. The chord (straight line across the opening) is also 9.549 m, and the sagitta (height of the arch above the chord midpoint) is 1.281 m.

Common Mistakes and Pitfalls

• Using degrees in the formula without converting – always convert to radians when using s = rθ. The calculator handles this for you.
• Mixing units – if the arc length is in meters, the radius will be in meters. Don’t mix feet and inches.
• Confusing chord and arc – the chord is a straight line; the arc is the curved part. They are equal only when the angle is 0° (degenerate) or 60° (as in the example, coincidentally).
• Rounding too early – if you round the radius to 9.5 m before computing the chord, you get 9.5 m instead of 9.549 m. Keep full precision until the final step.
• Assuming the angle is the interior angle of a polygon – the central angle is measured at the circle’s center, not at a vertex on the circle.

Limitations of the Formula

The formulas assume a perfect circle. Real-world objects may have slight irregularities, so measurements are approximate. Also, the angle must be between 0° and 360° (or 0 and 2π radians). For angles greater than 360°, the arc length repeats, but the formula still works if you reduce the angle modulo 360°. The sagitta formula is only valid for arcs less than a semicircle (θ < 180°); for larger arcs, the sagitta is measured differently (as the distance from the chord to the circle on the opposite side).

Frequently Asked Questions

How do I find the radius if I only know the chord length and sagitta?

Use the formula r = (c² / (8h)) + (h/2), where c is chord length and h is sagitta. For example, if c = 10 m and h = 2 m, then r = (100 / 16) + 1 = 6.25 + 1 = 7.25 m.

Can I calculate arc length without knowing the radius?

Yes, if you know the chord length and the sagitta. First compute the radius using the formula above, then find the central angle using θ = 2 × arcsin(c / (2r)) (in radians), and finally arc length = r × θ.

What is the difference between arc length and chord length?

Arc length is the distance along the curved line of the circle. Chord length is the straight-line distance between the two endpoints of the arc. The arc is always longer than the chord (except for a degenerate arc of 0°).

Why does the calculator use degrees instead of radians?

Most people measure angles in degrees in everyday contexts. The calculator converts degrees to radians internally, so you don’t have to. If you prefer radians, you can enter the angle in radians and multiply by 180/π, but the calculator expects degrees.

What if my angle is greater than 180°?

The formulas for chord and sagitta assume the arc is less than a semicircle (θ < 180°). For larger arcs, the chord length is still 2r sin(θ/2), but the sagitta becomes r + r cos(θ/2) (the distance from the chord to the circle on the opposite side). The arc length formula works for any angle up to 360°.

How accurate are the results?

The calculator gives results to three decimal places, which is precise enough for most practical purposes. For high-precision engineering, use more significant figures and avoid rounding intermediate steps.

Related

• Arc Length of Vector Calculator: Formula & Steps
• Arc Length to Straight Line Calculator
• Arc Length to Radius Calculator: Formula & Steps

in a | Arc length | m | 10 | 0
in theta | Central angle (degrees) | deg | 60 | 0
out r | Radius | m | (a * 180) / (theta * pi) | 3
out arc | Arc length | m | a | 3
out chord | Chord length | m | 2 * r * sind(theta / 2) | 3
out sag | Sagitta | m | r - r * cosd(theta / 2) | 3