Degree of Curve Calculator: Formula & Examples

This calculator works out the degree of curve for a circular curve, which is the central angle subtended by a 100-foot (or 100-meter) arc. You enter the radius of the curve, and the calculator returns the degree of curve in both arc definition and chord definition, plus the central angle for a given arc length. The result is used by surveyors and road designers to describe how sharp a horizontal curve is.

Formula

The arc definition of degree of curve (D) is the central angle that subtends an arc of 100 ft (or 100 m). The formula is:

D = (5729.578 / R) degrees

where R is the radius of the curve in feet (or meters). The constant 5729.578 is derived from 180 × 100 / π. For a 100-meter arc, the constant is 5729.578 as well (since 100 m / π × 180 = 5729.578 m-degrees).

The chord definition (used mainly in railway practice in the US) defines D as the central angle that subtends a chord of 100 ft. The formula is:

D = 2 × arcsin(50 / R)

where R is the radius in the same units as the chord length (50 ft is half the 100-ft chord).

Worked Example

Using the default radius of 500 ft:

• Arc definition: D = 5729.578 / 500 = 11.459° (rounded to 11.459°)
• Chord definition: D = 2 × arcsin(50 / 500) = 2 × arcsin(0.1) = 2 × 5.739° = 11.478°

The calculator shows 11.459° for the arc definition and 11.478° for the chord definition. The difference is small for flat curves but grows as the radius decreases.

When to Use the Degree of Curve

Degree of curve is a traditional way to specify horizontal alignment in highway and railway design. It is used in:

• Road centerline layout – to set out circular curves on plans and in the field.
• Railway track design – where the chord definition is still common in the US.
• Land surveying – to describe property boundaries that follow arcs.
• Civil engineering education – to compare curve sharpness without stating radius.

Many modern design software packages use radius directly, but degree of curve remains in older plans, specifications, and right-of-way documents. Knowing how to convert between the two is essential for reading legacy drawings.

How to Choose Between Arc and Chord Definitions

The choice depends on the context:

• Highway design (in the US and many other countries) uses the arc definition – D is based on a 100-ft arc length.
• Railway practice in the US historically uses the chord definition – D is based on a 100-ft chord.
• Metric countries often use a 100-meter arc or chord, but the same formulas apply with 100 m substituted.

If you are working from an old plan, check the legend or specification to see which definition was used. The difference is negligible for large radii (over 1000 ft) but can be several tenths of a degree for sharp curves.

Common Mistakes and How to Avoid Them

• Mixing units: The constant 5729.578 is only valid when the radius is in the same units as the arc length (feet or meters). If you measure the radius in meters but the arc is 100 ft, you must convert.
• Using the wrong formula: The chord formula uses arcsin, not the simple ratio. For a 500-ft radius, the difference is about 0.02°, but for a 100-ft radius the difference is over 2°.
• Forgetting that D is per 100 units: A degree of curve of 5° means the central angle for a 100-ft arc is 5°, not the total angle of the curve. The total central angle (Δ) is D × (arc length / 100).
• Rounding too early: If you round the constant to 5730, you introduce a small error. Use the full constant for precise work.

Limitations of the Formula

The degree of curve formula is exact only for a perfect circular arc. Real-world curves may be spirals (clothoids) that transition from tangent to circular, and the degree of curve changes along the spiral. The formula also assumes a flat plane; on a sphere (for very long curves) the geodesic curvature differs. For typical highway and railway curves, the circular approximation is sufficient.

Another limitation: the chord definition is not a simple linear relationship with radius. For very sharp curves (radius less than 50 ft), the chord formula gives a degree greater than 60°, which is impractical in design. Always check that your radius is within a reasonable range for the application.

Safety and Accuracy in Field Use

When setting out a curve in the field, the degree of curve alone is not enough – you also need the intersection angle (Δ), the tangent length, and the stationing of the PC (point of curvature) and PT (point of tangency). Mistakes in degree of curve calculations lead to misaligned roads or railways, which can cause unsafe driving conditions or derailments. Always double-check your inputs and use the correct definition (arc vs. chord) for the jurisdiction and project type. For critical work, have a second person verify the calculations.

Frequently Asked Questions

What is the difference between degree of curve and radius?

Degree of curve is the central angle subtended by a standard arc (or chord) of 100 units. Radius is the actual radius of the circle. They are inversely related: a larger radius gives a smaller degree of curve (a flatter curve). The conversion is D = 5729.578 / R for arc definition.

Why are there two definitions of degree of curve?

Historically, railroads used the chord definition because it was easier to measure chords in the field with a tape. Highways later adopted the arc definition because it is mathematically more direct for computing lengths along the curve. Both are still in use, so you need to know which one a plan uses.

Can I use this calculator for metric curves?

Yes, if you enter the radius in meters and use a 100-meter arc or chord. The constant 5729.578 works for meters as well because it is derived from 100 × 180 / π. For a 100-meter chord, the half-chord length is 50 meters, so the chord formula also works with meters.

What is the degree of curve for a 1000-ft radius?

Using the arc definition: D = 5729.578 / 1000 = 5.730°. Using the chord definition: D = 2 × arcsin(50/1000) = 2 × arcsin(0.05) = 2 × 2.866° = 5.732°. The difference is only 0.002°.

How do I find the length of a curve if I know the degree of curve and the central angle?

For the arc definition, the length of the curve (L) is L = (Δ / D) × 100, where Δ is the central angle of the full curve in degrees. For example, if Δ = 30° and D = 5°, then L = (30/5) × 100 = 600 ft.

Is degree of curve the same as the angle of intersection?

No. The angle of intersection (Δ) is the total change in direction between the two tangents. The degree of curve (D) is the central angle per 100 units of arc or chord. The total central angle of the curve equals Δ, and the curve length is proportional to Δ/D.

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in r | Radius | ft | 500 | 0
out arc_deg | Degree of curve (arc) | degrees | (5729.578 / r) * (180 / pi) | 3
out chord_deg | Degree of curve (chord) | degrees | 2 * asind(50 / r) | 3
out central_angle | Central angle (per 100 ft) | degrees | (5729.578 / r) * (180 / pi) | 3