Horizontal Curve Calculator Online

This calculator computes the key geometric elements of a simple circular horizontal curve—tangent length, external distance, middle ordinate, and long chord—from the curve’s radius and intersection angle (delta). You enter the radius and the delta angle in degrees, and the calculator returns the tangent length, external distance, middle ordinate, and long chord, all in the same unit you used for the radius.

Formula

The formulas for a simple circular horizontal curve are standard in highway and railway design. Given the radius R (in any length unit) and the intersection angle Δ (in degrees), the elements are:

• Tangent length (T): T = R × tan(Δ / 2)
• External distance (E): E = R × (sec(Δ / 2) − 1) = R × (1 / cos(Δ / 2) − 1)
• Middle ordinate (M): M = R × (1 − cos(Δ / 2))
• Long chord (LC): LC = 2 × R × sin(Δ / 2)

All results are in the same unit as the radius (e.g., meters, feet). The delta angle is the angle between the two tangents at the point of intersection (PI), also called the intersection angle or central angle.

Worked Example

Using the default values from the calculator: radius R = 300 m, delta Δ = 30°.

1. Tangent length: T = 300 × tan(15°) = 300 × 0.267949 = 80.38 m
2. External distance: E = 300 × (1 / cos(15°) − 1) = 300 × (1.035276 − 1) = 300 × 0.035276 = 10.58 m
3. Middle ordinate: M = 300 × (1 − cos(15°)) = 300 × (1 − 0.965926) = 300 × 0.034074 = 10.22 m
4. Long chord: LC = 2 × 300 × sin(15°) = 600 × 0.258819 = 155.29 m

These match the calculator output (rounded to two decimals).

When to Use a Horizontal Curve Calculator

Horizontal curve calculators are used in road design, railway alignment, and site planning. They help engineers and surveyors lay out the geometry of a curve that connects two straight sections (tangents) of a route. Typical uses include:

• Determining the length of curve needed for a given radius and deflection angle.
• Setting out the curve in the field using tangent and chord measurements.
• Checking whether a proposed curve meets design standards for sight distance or superelevation.
• Estimating right-of-way widths or clearances from the external distance.

The calculator is also useful for students learning route surveying or transportation engineering.

How to Read the Results

The tangent length tells you how far back from the point of intersection (PI) the curve begins (point of curvature, PC) and ends (point of tangency, PT). The external distance is the distance from the PI to the midpoint of the curve—useful for checking clearance at the PI. The middle ordinate is the distance from the midpoint of the long chord to the arc, which matters for sight distance. The long chord is the straight-line distance between PC and PT.

All values are in the same unit you entered for the radius. If you enter the radius in feet, all outputs are in feet. For metric, use meters or kilometers, but be consistent.

Common Mistakes and Misconceptions

• Using the wrong angle mode: The formulas require the delta angle in degrees. If your calculator is in radians, convert degrees to radians (multiply by π/180) before using trig functions.
• Confusing delta with the deflection angle at PC: The deflection angle to a point on the curve is Δ/2 for the midpoint, but the formulas above use Δ directly.
• Forgetting that T, E, M, LC are all in the same unit as R: If you mix meters and feet, results will be wrong.
• Assuming the curve is a full circle: The formulas apply only to a simple circular arc. Spiral (clothoid) curves require different calculations.

A common misconception is that the external distance equals the middle ordinate—they are close for small angles but not equal. For a 30° curve, E ≈ 10.58 m and M ≈ 10.22 m, a difference of 0.36 m.

Limitations of the Simple Circular Curve Model

This calculator assumes a simple circular curve with no transition spirals. Real highways often use spiral curves to ease the transition from tangent to circular arc, especially at higher speeds. The simple curve formulas still provide a good first approximation, but for final design, you may need to account for superelevation, sight distance, and spiral length.

Also, the radius in practice is often chosen based on design speed and superelevation rate, not arbitrarily. The calculator gives geometry only; it does not tell you what radius is safe for a given speed.

Alternatives and Related Calculations

If you need to compute the actual arc length of the curve, use: L = R × Δ (in radians) = R × Δ × (π/180). For example, with R = 300 m and Δ = 30°, L = 300 × 30 × (π/180) = 157.08 m.

For a full stationing layout, you might also need the chord length to any point, which involves the deflection angle from the tangent. That is beyond the scope of this simple calculator but is covered in surveying textbooks.

Frequently Asked Questions

What is the difference between the external distance and the middle ordinate?

The external distance is measured from the point of intersection (PI) to the arc, along the line that bisects the angle. The middle ordinate is measured from the midpoint of the long chord to the arc, perpendicular to the chord. For a given curve, the external distance is always larger than the middle ordinate, but the difference is small for gentle curves.

Can I use this calculator for railway curves?

Yes, the geometry is the same. However, railways often use larger radii and may require special clearance checks. The calculator gives the basic geometric elements, but you may need additional calculations for vehicle clearance and superelevation.

What units should I use for the radius?

Any length unit works, as long as you are consistent. If you enter the radius in feet, all outputs are in feet. If you enter in meters, outputs are in meters. The delta angle is always in degrees.

How do I find the radius if I know the tangent length and delta?

Rearrange the tangent formula: R = T / tan(Δ/2). For example, if T = 80.38 m and Δ = 30°, then R = 80.38 / tan(15°) = 80.38 / 0.267949 = 300 m.

What is the point of curvature (PC) and point of tangency (PT)?

PC is the point where the curve begins (the start of the arc), and PT is where it ends. The tangent length measured from the PI back to the PC and forward to the PT defines these locations along the alignment.

Does the calculator account for superelevation?

No. Superelevation is the banking of the road on a curve, which is a separate design consideration that depends on design speed and friction. The calculator only computes geometric elements of the curve itself.

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in r | Radius | m | 300 | 0
in d | Delta (degrees) | deg | 30 | 0
out t | Tangent length | m | r * tand(d / 2) | 2
out e | External distance | m | r * (1 / cosd(d / 2) - 1) | 2
out m | Middle ordinate | m | r * (1 - cosd(d / 2)) | 2
out lc | Long chord | m | 2 * r * sind(d / 2) | 2