Turnout Design Calculator
Calculate railroad switch/turnout geometry from frog number — frog angle, lead length, curve radius, and maximum diverging speed.
About this calculator
A railroad turnout's geometry is defined almost entirely by its frog number N — the ratio describing how gradually the diverging route splits from the main track. This calculator derives the frog angle exactly as θ = 2·arctan(1/(2N)), then uses standard gauge (56.5 in) to compute the lead length (switch point to theoretical frog nose) as N times the gauge in feet, and approximates the diverging route's curve radius as R ≈ G / (2·sin²(θ/2)) — the standard small-angle relationship between a turnout's frog number and the radius of the curve it produces. From that radius it derives the equivalent degree of curve (5729.578/R, the standard railway curve-degree conversion) and then the maximum safe diverging-route speed using the AREMA superelevation formula e = 0.0007·D·V², solved for V using your entered maximum unbalanced cant (default 1.5 inches, the typical turnout limit).
Higher frog numbers (like #15 or #20, used on main lines) produce gentler angles, larger radii, and higher permissible diverging speeds; lower frog numbers (like #6, common in yards) are sharper and slower but more compact. The heel spread and closure rail length are secondary approximations useful for construction planning, not AREMA-certified dimensions. This tool is meant for conceptual design and estimating — real turnout construction must follow the exact AREMA plans and manufacturer standards for the specific frog number, rail section, and speed class involved, not the approximate formulas used here.
Inputs
Results
Frog angle
5.73°
Figures current as of 2026. Source: American Railway Engineering and Maintenance-of-Way Association (AREMA), Manual for Railway Engineering, Chapter 5: Track
How to Use This Calculator
- Enter Frog number (N) and Max unbalanced cant.
- Review the Frog angle (°) result.
- Use Turnout curve radius (ft) and Degree of curve (°) to inform your decision.
How the result changes with Frog number (N)
| Frog number (N) | Frog angle |
|---|---|
| 5 | 11.42° |
| 7.5 | 7.63° |
| 15 | 3.82° |
| 25 | 2.29° |
What each input means
- Frog number (N)
- Turnout frog number (#6=yard, #10=branch, #15=main, #20=high-speed).
- Max unbalanced cant
- Maximum allowed cant deficiency through the turnout (typically 1.5 in).
What each result means
- Frog angle
- Angle of divergence at the frog point.
- Turnout curve radius
- Radius of the diverging route through the turnout.
- Degree of curve
- Equivalent degree of curvature.
- Lead length
- Distance from switch point to theoretical frog nose.
- Overall turnout length
- Approximate total length of the turnout assembly.
- Max diverging speed
- Maximum safe speed through the diverging route.
- Heel spread
- Spread between stock rail and switch rail at the heel block.
- Closure rail length
- Approximate length of the closure rail section.
- Speed class
- 0=yard/low, 1=medium, 2=main line, 3=high-speed.
How this is calculated
Worked example, using the default values
- Identify Input ParametersFrog number (N) = 10, Max unbalanced cant = 1.5 = 2 input(s) provided
- Calculate Frog angleFrog angle = frogAngleRad * (180 / π)5.725 = 5.725
- Calculate Turnout curve radius944 = 944
- Calculate Degree of curveDegree of curve = 5729.578 / turnoutRadiusFt6.07 = 6.07
Figures and sources
- AREMA equilibrium superelevation (cant) formula applied to turnout diverging-route geometry (2026) — American Railway Engineering and Maintenance-of-Way Association (AREMA), Manual for Railway Engineering, Chapter 5: Track
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why does a higher frog number produce a gentler turnout?
Frog angle is computed as θ = 2·arctan(1/(2N)), so as frog number N increases, 1/(2N) shrinks toward zero and the angle shrinks with it — a #20 frog diverges far more gradually than a #6. That gentler angle also produces a larger radius in the R ≈ G/(2·sin²(θ/2)) approximation, which is why higher-numbered frogs support higher diverging speeds.
How does the calculator get from frog number to maximum diverging speed?
It first derives degree of curve as 5729.578 divided by the turnout radius, then applies the same equilibrium superelevation formula AREMA's Manual for Railway Engineering (Chapter 5, Track) publishes for curve design, e = 0.0007·D·V² — solved for V using your entered maximum unbalanced cant (default 1.5 inches) — to compute the fastest speed a train can take the diverging route without exceeding that cant limit.
What's the difference between lead length and overall turnout length?
Lead length is the distance from the switch point to the theoretical nose of the frog, computed as frog number times gauge in feet. Overall turnout length adds roughly half that same lead-length-scaled distance again (gaugeFt × frogNumber × 0.5) to approximate the additional length needed to complete the assembly, so it's always longer than the lead length alone.
Why does the calculator warn that heel spread and closure rail length aren't AREMA-certified dimensions?
Heel spread is computed as a simple ratio (gauge ÷ frog number) and closure rail length as 60% of lead length — both are quick geometric approximations useful for early planning, not values taken from actual AREMA turnout plans, which specify exact rail lengths, curve data, and hardware dimensions per frog number and rail section that this calculator doesn't model.
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