Curve Geometry Calculator: Radius & Arc Length

This curve geometry calculator works out the radius, arc length, and chord length of a circular curve when you enter the central angle and either the radius or the arc length. It is built for road and rail horizontal curves, circular arcs in CAD work, and any layout task where you need the missing dimension of a simple circular curve. You get the radius, arc length, chord length, and sagitta (mid-ordinate) in consistent units, so you can move straight from field measurements to a drawing or staking plan.

Formula

For a circular curve with central angle θ (in degrees) and radius R (in any length unit), the three main quantities are:

• Arc length L = (π / 180) × θ × R
• Chord length C = 2 × R × sin(θ / 2)
• Sagitta (mid-ordinate) M = R × (1 − cos(θ / 2))

Here θ is the angle subtended by the arc at the centre of the circle, R is the radius from the centre to the curve, L is the distance along the curve itself, C is the straight-line distance between the two endpoints of the arc, and M is the perpendicular distance from the midpoint of the chord to the midpoint of the arc. The formulas assume a perfectly circular arc and use the same length unit for all outputs (metres in the default, but any unit works as long as you are consistent).

Worked Example

Using the default values from the calculator: central angle θ = 60°, radius R = 100 m.

1. Convert the angle to radians: 60 × π / 180 = 1.0472 rad.
2. Arc length L = 1.0472 × 100 = 104.72 m.
3. Chord length C = 2 × 100 × sin(30°) = 200 × 0.5 = 100.00 m.
4. Sagitta M = 100 × (1 − cos(30°)) = 100 × (1 − 0.8660) = 13.40 m.

The calculator gives the same results: arc length 104.720 m, chord 100.000 m, sagitta 13.397 m (rounded to 13.397). Notice that for a 60° arc the chord and the radius happen to be equal, which is a useful sanity check.

When to Use a Curve Geometry Calculator

This tool is for any situation where you have a circular arc and need one of its geometric properties. Common uses include:

• Laying out horizontal curves for roads, railways, or sidewalks.
• Computing arc lengths for pipe bends, conveyor curves, or cable trays.
• Checking the geometry of a circular segment in woodworking or metal fabrication.
• Converting between radius and degree of curvature (where degree of curve is the central angle subtended by a 100-ft or 30-m chord, depending on the convention).

If you only know the chord length and the sagitta (for example, from measuring an existing arch), you can rearrange the sagitta formula to solve for the radius: R = (C² / (8M)) + (M / 2), where C is the chord and M is the sagitta. That is a common field method for measuring an existing curve without a total station.

How to Read the Results

The first output, arc length, is the distance along the curve itself. This is what you would paint on the ground or measure with a tape along the curve. The chord length is the straight-line distance from the start to the end of the arc, which is what you would measure if you pulled a tape tight between the two endpoints. The sagitta is the distance from the midpoint of the chord to the midpoint of the arc; it tells you how much the curve bulges away from the straight line.

All outputs are in the same unit you entered for the radius. If you enter the radius in feet, the arc length, chord, and sagitta are all in feet. If you enter it in metres, all outputs are in metres. The central angle is always in degrees; if your angle is in radians or grads, convert it first (multiply radians by 180/π, or grads by 0.9 to get degrees).

Common Mistakes and Misconceptions

The most frequent error is mixing up the central angle with the deflection angle. The central angle is the angle at the centre of the circle between the two radii that define the arc. The deflection angle in surveying is the angle between the tangent at the start of the curve and the chord to a point on the curve; it is always half the central angle for a given point. For a full arc, the deflection angle to the end point is θ/2, not θ.

Another common mistake is using degrees in a formula that expects radians. Many calculators and software tools default to radians, so always check the mode. The formulas here use degrees directly, but the conversion to radians is built in.

A third issue is confusing the sagitta with the versine. They are the same thing for a circular arc. The versine is defined as 1 − cos(θ), and the sagitta is R × (1 − cos(θ/2)). For a full circle, the sagitta equals the diameter, which makes sense.

Finally, remember that these formulas apply only to circular arcs. If your curve is a spiral (clothoid) or a parabola, these formulas do not apply; you need the specific equations for that curve type.

Limits of the Formula

The formulas are exact for a perfect circular arc in a plane. They do not account for:

• Earth curvature or projection distortions on large survey areas.
• Vertical curves, which are usually parabolas, not circular arcs.
• Transition spirals that are often inserted between a tangent and a circular curve on highways and railways.
• Non-circular shapes like ellipses or clothoids.

For most construction and design work, the circular arc is accurate enough. For very large curves (radius over a few kilometres), the difference between the chord and the arc length becomes negligible for short arcs, but the formulas still hold exactly.

How to Choose the Right Input

You need two independent quantities to define a circular arc: the central angle and the radius. If you have the arc length and the radius, you can compute the central angle as θ = (180 / π) × (L / R). If you have the chord length and the radius, use θ = 2 × arcsin(C / (2R)). If you have the chord and the sagitta, use the formula given earlier to find the radius first, then compute the angle.

In practice, surveyors often measure the chord and the sagitta directly in the field because they are easy to obtain with a tape and a level. The radius can then be derived. This calculator works the other way: you input the radius and angle, and get the chord and sagitta. If you need to go from chord and sagitta to radius, you can use a separate calculator or the rearranged formula.

Frequently Asked Questions

What is the difference between arc length and chord length?

Arc length is the distance measured along the curved line itself, like driving along the curve. Chord length is the straight-line distance between the two endpoints, like flying from one end to the other. For a given radius and angle, the chord is always shorter than the arc, except for a zero angle where they are equal.

Can I use this calculator for a full circle?

Yes, set the central angle to 360 degrees. The arc length becomes the circumference (2πR), the chord length becomes 0 (because the endpoints coincide), and the sagitta becomes 2R (the diameter). The calculator will show the chord as 0, which is correct.

What units should I use?

Any consistent length unit works for the radius. If you enter the radius in feet, all outputs are in feet. If you enter it in metres, outputs are in metres. The angle is always in degrees. There is no unit conversion needed as long as you use the same unit for the radius.

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

Use the formula R = (C² / (8M)) + (M / 2), where C is the chord length and M is the sagitta. For example, if C = 10 m and M = 1 m, then R = (100 / 8) + 0.5 = 12.5 + 0.5 = 13 m. Then you can compute the central angle using θ = 2 × arcsin(C / (2R)).

Why does the chord length come out as 0 for a 180-degree arc?

For a semicircle, the chord is the diameter, which is 2R, not 0. If you enter 180 degrees and R = 100 m, the chord is 2 × 100 × sin(90°) = 200 m. The chord is 0 only for a full 360-degree arc, because the endpoints are the same point.

Is this the same as a curve calculator for roads?

Yes, road and railway horizontal curves are typically circular arcs. This calculator gives the basic geometry. In real design, you also need to consider superelevation, sight distance, and transition spirals, but the pure circular curve geometry is exactly what this calculator provides.

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in a | Central angle | degrees | 60 | 0
in r | Radius | m | 100 | 0
out arc | Arc length | m | (a * pi / 180) * r | 3
out chord | Chord length | m | 2 * r * sin(a * pi / 360) | 3
out sag | Sagitta (mid-ordinate) | m | r * (1 - cos(a * pi / 360)) | 3