Circular Curve Calculator: Radius, Length, and More

A circular curve calculator works out the key dimensions of a simple circular arc—radius, arc length, chord length, central angle, and sagitta—from any two known values you enter. You provide two measurements (like radius and central angle, or chord and sagitta), and the calculator returns the rest, using standard geometry formulas. This is useful for road and railway design, surveying, woodworking, and any project involving a circular arc.

Formula

The calculator uses four basic circular curve formulas. All angles are in degrees, and lengths are in the same unit you enter (meters, feet, etc.).

• Arc length (L): L = r × (a × π / 180), where r is the radius and a is the central angle in degrees. This is the distance along the curve itself.
• Chord length (C): C = 2 × r × sin(a / 2). The chord is the straight-line distance between the two endpoints of the arc.
• Sagitta (M) (also called mid-ordinate): M = r × (1 − cos(a / 2)). This is the perpendicular distance from the midpoint of the chord to the arc.
• Tangent length (T): T = r × tan(a / 2). In road design, this is the distance from the point of intersection (PI) to the point of curvature (PC) along a straight tangent.

All variables are defined as: r = radius (same unit as lengths), a = central angle (degrees), L = arc length, C = chord length, M = sagitta, T = tangent length.

Worked Example

Using the default values from the calculator above: radius r = 100 m and central angle a = 45°.

1. Arc length: L = 100 × (45 × π / 180) = 100 × 0.7854 = 78.54 m.
2. Chord length: C = 2 × 100 × sin(22.5°) = 200 × 0.38268 = 76.54 m.
3. Sagitta: M = 100 × (1 − cos(22.5°)) = 100 × (1 − 0.92388) = 7.61 m.
4. Tangent length: T = 100 × tan(22.5°) = 100 × 0.41421 = 41.42 m.

These match the calculator’s outputs (rounded to two decimals). Notice the arc length (78.54 m) is slightly longer than the chord (76.54 m), as you would expect for a curved path.

When to Use a Circular Curve Calculator

This tool is most often used in:

• Road and railway design: Engineers use circular curves to connect straight sections of road or track. The tangent length and sagitta help set out the curve in the field.
• Surveying: Land surveyors stake out curve centerlines using chord and tangent measurements.
• Woodworking and metalworking: When building arched frames, curved rails, or circular stairs, you need the radius and arc length to cut materials accurately.
• Landscaping: Designing curved paths, flower beds, or retaining walls with a specific arc.
• Architecture: For arched windows, doorways, or decorative elements.

You might also use it in reverse: if you know the chord length and sagitta (which you can measure directly on an existing curve), you can solve for the radius using the formula r = (C² / (8M)) + (M / 2). That is a common practical application when you cannot access the center of the circle.

How to Choose the Right Inputs

The calculator requires exactly two inputs. The most common pairs are:

• Radius and central angle: The most straightforward. You know the curve’s sharpness and how much of the circle you need.
• Chord and sagitta: Useful when you have an existing arc and can measure the straight distance across it and the depth at the midpoint.
• Radius and arc length: If you know the radius and how long the curve is, you can find the angle (a = (L / r) × (180 / π)).

If you only have one measurement, you cannot determine the curve uniquely—there are infinitely many circles that pass through a single point. You need at least two independent measurements.

Common Mistakes and How to Avoid Them

• Mixing degrees and radians: The calculator uses degrees. If you enter a central angle in radians (like 0.7854 instead of 45), all results will be wrong. Always convert radians to degrees by multiplying by 180/π.
• Using the wrong radius: In road design, the radius is measured to the centerline of the road, not the inner or outer edge. For a curb or rail, the radius may be to the inside edge. Be consistent.
• Forgetting the sagitta is not the same as the rise: The sagitta is measured perpendicular to the chord, not along the radius. If you measure the rise at the midpoint, that is the sagitta only if the chord is horizontal.
• Rounding too early: If you round intermediate values (like sin(22.5°)) to a few decimals, your final answer may be off. Keep full precision until the final step.

Limitations of the Formula

These formulas assume a perfect circular arc. Real-world curves may be spirals (like highway transition curves) or compound curves with multiple radii. For those, you need more advanced calculations. Also, the formulas give the geometric dimensions, not the actual construction layout—on site, you would use surveying instruments to set out points along the curve, often at regular intervals (e.g., every 10 m). The calculator gives you the overall dimensions, but not the coordinates of intermediate points.

Another limitation: if the central angle is very large (over 180°), the chord length formula still works, but the sagitta formula gives the distance from the chord to the arc on the side opposite the center. For a full circle (a = 360°), the chord length becomes 0 and the sagitta equals the radius. That is correct, but you may need to interpret the results carefully.

Frequently Asked Questions

What is the difference between arc length and chord length?

Arc length is the distance along the curved path itself, like driving around a bend. Chord length is the straight-line distance between the two endpoints, like a bird flying across the curve. The arc length is always longer than the chord (except when the angle is 0°).

How do I find the radius if I only know the chord and 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. This is handy when you cannot measure to the center of the circle.

What is the tangent length used for?

In road and railway design, the tangent length is the distance from the point where the straight road ends (point of curvature) to the point where the two tangents would intersect (point of intersection). It tells surveyors where to start and end the curve along the straight sections.

Can I use this calculator for a full circle?

Yes, but set the central angle to 360°. The arc length will be the circumference (2πr), the chord length will be 0, and the sagitta will equal the radius. For a full circle, the chord and tangent are not meaningful, so you would just use the arc length.

What units should I use?

Any consistent unit works—meters, feet, inches, etc. The calculator does not care, as long as you use the same unit for all lengths. The central angle is always in degrees. If you want results in both metric and imperial, convert the final numbers using 1 m = 3.28084 ft.

Why does the calculator ask for the central angle in degrees, not radians?

Degrees are more common in surveying and construction. Most people think of a curve as “a 45° curve” rather than “0.7854 radians.” The calculator converts degrees to radians internally for the formulas, so you do not have to.

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in r | Radius | m | 100 | 0
in a | Central angle | degrees | 45 | 0
out L | Arc length | m | r * (a * pi / 180) | 2
out C | Chord length | m | 2 * r * sind(a / 2) | 2
out M | Sagitta (mid-ordinate) | m | r * (1 - cosd(a / 2)) | 2
out T | Tangent length | m | r * tand(a / 2) | 2