Curve Calculator Radius: Find the Radius

A curve calculator radius tool works out the radius of a circular curve from measurements you already have—most commonly the chord length and the sagitta (the middle ordinate), or the arc length and central angle. You enter those two known values, and the calculator returns the radius in the same unit you used for the inputs, along with related values like the degree of curvature. This is the same geometry used by road designers, rail engineers, machinists, and woodworkers when they need to reproduce or check a curve without a full drawing.

Formula

The calculator uses the standard circular segment formula that relates chord length, sagitta, and radius:

R = (C² / (8 × s)) + (s / 2)

Where:

• R = radius of the circle (same unit as C and s)
• C = chord length – the straight-line distance between the two endpoints of the arc
• s = sagitta – the perpendicular distance from the midpoint of the chord to the arc (also called the middle ordinate)

This formula comes from the Pythagorean theorem applied to the right triangle formed by the radius, half the chord, and the radius minus the sagitta. It is exact for a circular arc, not an approximation.

Worked Example

Using the default values from the calculator: chord length C = 10 m, sagitta s = 1 m.

1. Square the chord: 10² = 100
2. Multiply sagitta by 8: 8 × 1 = 8
3. Divide: 100 / 8 = 12.5
4. Add sagitta/2: 12.5 + 0.5 = 13.0

So the radius is 13.0 m (rounded to 3 decimals). The degree of curvature (arc definition, used in road design) is about 44.0°.

When to Use This Calculator

You reach for this calculation in any situation where you have a curved shape but no drawing or CAD file with the radius labelled. Common real-world cases:

• Road and railway design: surveyors measure the chord and middle ordinate of an existing curve to find its radius and check it against design standards.
• Machining and metalwork: you need to cut a curved piece to match an existing template or a bent part.
• Woodworking and carpentry: building an arched doorway, curved cabinet face, or bent lamination requires knowing the radius of the form.
• Landscape and hardscape: laying out a curved garden bed, path, or retaining wall from two stakes and a string.
• Restoration work: reproducing a curved trim or arch from an old piece where the radius is not marked.

The chord–sagitta method is especially useful because both measurements are easy to take on site with a tape measure and a straightedge.

How to Measure the Inputs

Getting accurate inputs is the difference between a useful answer and a misleading one. Here’s how to measure each:

Chord length (C)

Pick two points on the curve that you can see and reach. The chord is the straight line between them—not the distance along the curve. Use a tape measure pulled tight. The longer the chord, the more accurate the final radius, but the sagitta must still be measurable. A common approach is to use the full width of the curve if it has clear endpoints.

Sagitta (s)

Find the midpoint of the chord. From that midpoint, measure the perpendicular distance up to the curve. A framing square helps ensure the measurement is perpendicular. If the curve is large, use a string stretched between the chord endpoints and measure from the string’s midpoint to the curve.

For a small arc, you can also use a digital caliper or a depth gauge. The more precisely you measure the sagitta, the more accurate the radius—small errors in the sagitta cause large errors in the radius, especially when the sagitta is small relative to the chord.

Common Mistakes and How to Avoid Them

• Mixing units: both inputs must be in the same unit. If you measure the chord in feet and the sagitta in inches, convert one first. The calculator assumes the same unit for both.
• Measuring the arc instead of the chord: the formula needs the straight-line chord, not the curved arc length. Using arc length by mistake gives a wrong radius.
• Not measuring sagitta at the true midpoint: if you measure at any other point along the chord, the formula fails. Mark the midpoint carefully.
• Sagitta too small to measure accurately: for a very flat curve, the sagitta might be a few millimetres. A tape measure won’t give you the precision you need; use a caliper or a machinist’s rule.
• Rounding too early: if you round the sagitta to one decimal before calculating, the radius can be off by a lot. Keep full precision until the final answer.

Limits of the Chord–Sagitta Formula

This formula works only for a circular arc. If the curve is elliptical, spiral, or irregular (like a road transition curve), the result will be an approximation at best. For a spiral curve, the radius changes along its length, so a single value is meaningless.

Also, the formula is most stable when the sagitta is a reasonable fraction of the chord. If the sagitta is extremely small relative to the chord, tiny measurement errors dominate and the computed radius can swing wildly. In such cases, measure a longer chord or use a different method, such as three points on the curve with a coordinate geometry solution.

Finally, the degree of curvature output uses the arc definition (central angle subtending a 100-unit arc), which is common in road design in the US. Railway engineers in some regions use the chord definition (central angle subtending a 100-unit chord). The two differ slightly; the calculator gives the arc definition.

Alternatives to the Chord–Sagitta Method

If you can’t measure a chord and sagitta easily, other approaches exist:

• Three-point method: measure the coordinates of three points on the curve. The radius is the circumradius of the triangle formed. This is more field work but works on partial curves.
• Arc length and central angle: if you know the arc length (L) and the central angle (θ in degrees), radius R = (180 × L) / (π × θ).
• Tangent offset method: measure offsets from a tangent line at known distances. This is a surveying technique for large curves.
• CAD or photo software: if you have a scaled drawing or a photo with a known dimension, you can fit a circle in software and read the radius.
• Radius gauge or template: for small curves, a set of radius gauges lets you physically match the curve.

Each method has its own accuracy and practicality. The chord–sagitta method is usually the best balance of simple measurement and reliable math for field work.

Frequently Asked Questions

Can I use this calculator for a full circle?

No. For a full circle, the chord and sagitta are not defined in the same way. If you have a full circle, measure the diameter directly (the widest distance across) and divide by 2 to get the radius. The chord–sagitta formula is for an arc that is less than a full circle.

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

That combination does not have a closed-form solution for the radius. You would need to solve a transcendental equation numerically, which is beyond this simple calculator. You can use an iterative method or a dedicated solver that handles arc length and chord length.

How accurate is the radius from this calculator?

It is mathematically exact for a circular arc given perfect inputs. In practice, accuracy depends entirely on how well you measure the chord and sagitta. A 1% error in the sagitta typically causes roughly a 1% error in the radius, but the relationship is not linear—for very small sagittas, the error amplifies.

What is the degree of curvature used for?

In road and rail design, the degree of curvature (D) describes how sharp a curve is. It is the central angle subtended by a 100-unit arc (arc definition). A larger D means a sharper curve (smaller radius). It is used in design tables and speed calculations, though modern design increasingly uses radius directly.

Can I use feet and inches together?

Only if you convert them to a single unit first. For example, convert feet to inches (multiply by 12) or inches to feet (divide by 12). The calculator has no way to know that 10 and 6 are meant as 10 feet 6 inches—it treats them as 10 and 6 in the same unit.

Why does my measured radius differ from the drawing?

Possible reasons: the curve is not perfectly circular (it may be a spiral or an ellipse), your measurements have small errors, or the drawing uses a different definition of radius (for example, the radius to the centerline vs. the radius to the edge of a road). Check the drawing’s notes and re-measure the sagitta carefully.

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in chord | Chord length (C) | m | 10 | 0
in sagitta | Sagitta (s) | m | 1 | 0
out radius | Radius (R) | m | (chord^2 / (8 * sagitta)) + (sagitta / 2) | 3
out degree | Degree of curvature (D) | degrees | 2 * asind(chord / (2 * ((chord^2 / (8 * sagitta)) + (sagitta / 2)))) | 3