Vertical alignment
Vertical Curve Calculator
Crest and sag curves in one tool. Solve for the required length, the K value, station elevations and the high or low point, or check the sight distance an existing curve actually provides — with every substitution shown and every constant traced to the manual it comes from.
Calculations run entirely in your browser. Your engineering inputs are never uploaded or stored.
A = G2 − G1 = -5.00% → crest curve. The grade change is negative, so the profile turns downward.
Not sure what sight distance to use? The stopping sight distance calculator derives it from design speed and grade, and can hand the value back here. The adopted design SSD at 50 mph is .
Minimum length of crest vertical curve
418.4 ft
- Required for sight distance
- 418.4 ft
- Governing case
- S ≥ L
- K
- 83.7
Minimum K at 50 mph: 83.70 computed, 84 adopted in TxDOT Table 4-12.
How this was calculated
Algebraic difference in grades
A = G2 − G1
A = -2 − (3)
= -5.00%
Required length (S ≥ L case)
L = 2S − 2158 / A
L = 2 × 425 − 2158 / 5.00
= 418.4 ft
Rate of vertical curvature
K = L / A
K = 418.4 / 5.00
= 83.7
Assumptions
| Quantity | Value | Source |
|---|---|---|
| Incoming grade G1 | 3% | User input |
| Outgoing grade G2 | -2% | User input |
| Sight distance | 425.0 ft | User input |
| Driver eye height | 3.5 ft | INDOT, Indiana Design Manual, Chapter 44 — Vertical Alignment — Chapter 44 — Vertical Alignment, 44-3.01 Crest Vertical Curve |
| Object height | 2 ft | INDOT, Indiana Design Manual, Chapter 44 — Vertical Alignment — Chapter 44 — Vertical Alignment, 44-3.01 Crest Vertical Curve |
- Driver eye height. A 5th percentile passenger car value. A truck driver sits higher, at about 7.6 ft, and the source notes that the higher eye height offsets the longer stopping distance a truck needs.
- Object height. Represents the taillights of a passenger car ahead. California uses a ½ ft object height for stopping sight distance instead, which produces longer required crest curves for the same sight distance.
References
Indiana Design Manual, Chapter 44 — Vertical Alignment
The four crest and sag vertical curve length equations for both S < L and S ≥ L, printed with their derivation: the crest constant 2158 as 100(√(2h₁)+√(2h₂))² for a 3.5 ft eye height and 2 ft object height, and the sag headlight criterion assuming a 2 ft headlight height and a 1 degree upward divergence of the beam.
Stopping sight distance equations for level and graded roadways, the 2.5 s brake reaction time and 11.2 ft/s² deceleration rate, and the calculated and adopted design SSD values in Tables 4-23 and 4-24.
About this method
A crest curve limits sight distance because the road surface itself blocks the view over the apex. The required length therefore depends on how sharply the grade changes (A) and how far ahead the driver must see (S).
Two equations are published because the geometry differs depending on whether the sight line stays within the curve. When the required length works out longer than the sight distance, the S < L form applies; when shorter, the S ≥ L form does. This calculator evaluates both and reports whichever case is self-consistent.
For a small enough grade change the sight line passes clear over the apex and sight distance imposes no length requirement at all. A minimum length rule governs the design instead — TxDOT uses three times the design speed, and is explicit that this is not a design control.
The minimum K values published in design tables are simply S² / 2158 evaluated at each design speed’s stopping sight distance, then rounded up. That is why this calculator and the stopping sight distance calculator agree: they are driven by the same numbers.
Limitations
- Applies to equal-tangent parabolic curves, which is the standard form in highway profile design.
- Assumes passenger car eye and object heights. Truck and decision sight distance criteria use different heights and produce different lengths.
- Sight distance is the governing criterion here. Drainage, comfort, appearance and vertical clearance may each demand a longer curve.
What a vertical curve does
A vertical curve provides a gradual transition between two tangent grades. Highway profile design uses a simple equal-tangent parabola, which has the convenient property that the rate of grade change is constant along the curve.
That property is what makes K useful. Since K = L / A, and therefore L = KA, a design table can publish one minimum K per design speed rather than tabulating every combination of grade change and length. Checking an existing curve is equally simple: compute its K and compare against the design value.
The sign of A = G2 − G1 decides the curve type. Negative A turns the profile downward and gives a crest; positive A turns it upward and gives a sag.
Why crest and sag use different equations
The two curve types are limited by completely different things, and this is the single most important idea in vertical alignment design.
On a crest, the road surface itself blocks the driver's view over the apex. The governing geometry is a sight line from a driver's eye, 3.5 ft up, to an object 2 ft high on the pavement. Those two heights produce the constant 2158 that appears in the crest equations — it is 100(√(2h₁) + √(2h₂))², not an arbitrary number.
On a sag, nothing blocks the view. In daylight, sight distance across a sag curve is essentially unrestricted. The design case is night: headlights point slightly upward, so as the road curves up beneath them the illuminated distance shortens. The governing assumptions become a 2 ft headlight height and a 1 degree upward beam divergence, which produce the 400 and 3.5 in the sag equations.
Because the constraints are unrelated, the required K values cross over. Below roughly 55 mph a sag curve needs a larger K than a crest at the same design speed; above it, the crest requirement overtakes the sag. That crossing is visible in the table below and is a consequence of the physics, not a quirk of the tabulation.
Minimum K values by design speed
Transcribed from TxDOT Roadway Design Manual Table 4-12. Every value in the crest and sag columns is reproduced by this calculator's equations driven by that design speed's stopping sight distance.
| Design speed | Design SSD | Crest K | Sag K | Preferable min L |
|---|---|---|---|---|
| 15 mph | 80 | 3 | 10 | 45 |
| 20 mph | 115 | 7 | 17 | 60 |
| 25 mph | 155 | 12 | 26 | 75 |
| 30 mph | 200 | 19 | 37 | 90 |
| 35 mph | 250 | 29 | 49 | 105 |
| 40 mph | 305 | 44 | 64 | 120 |
| 45 mph | 360 | 61 | 79 | 135 |
| 50 mph | 425 | 84 | 96 | 150 |
| 55 mph | 495 | 114 | 115 | 165 |
| 60 mph | 570 | 151 | 136 | 180 |
| 65 mph | 645 | 193 | 157 | 195 |
| 70 mph | 730 | 247 | 181 | 210 |
| 75 mph | 820 | 312 | 206 | 225 |
| 80 mph | 910 | 384 | 231 | 240 |
K in feet per percent; distances in feet. Values shown in amber exceed TxDOT's drainage threshold of K > 167, which the manual is careful to describe as a point beyond which drainage needs attention rather than a design maximum. The preferable minimum length of three times the design speed is explicitly not a design control — a design waiver is not required on that basis alone if stopping sight distance is met.
Working from design speed to curve length
Vertical curve design usually starts a step earlier than this page. The sight distance that drives the equations here is itself derived from design speed and grade:
- Design speed → stopping sight distance, using the perception-reaction and deceleration assumptions of the governing agency.
- Stopping sight distance → minimum crest or sag curve length, using the equations on this page.
- Length and grades → K, station elevations, and the high or low point for the profile and drainage design.
The stopping sight distance calculator can hand its result directly to this one, and this page shows you the value it received so the chain stays auditable rather than hidden.
Common questions
What is the K value of a vertical curve?
K is the rate of vertical curvature: the horizontal distance, in feet, needed to produce a 1 percent change in grade. It is defined as K = L / A, where L is the curve length and A is the algebraic difference in grades. Because L = KA, design tables can publish a single minimum K per design speed instead of tabulating every combination of A and L.
How do I calculate the minimum length of a crest vertical curve?
When the sight distance is shorter than the curve, L = A × S² / 2158. When it is longer, L = 2S − 2158 / A. The constant 2158 is 100(√(2h₁) + √(2h₂))² using a 3.5 ft driver eye height and a 2 ft object height. Evaluate both forms and use whichever case is self-consistent with its own assumption.
Why are the sag curve equations different from the crest equations?
On a crest curve the road surface itself blocks the view over the apex, so the controlling geometry involves the driver eye height and object height. On a sag curve the surface blocks nothing — during daylight sight distance is essentially unrestricted. The design case is night driving, where the headlight beam limits how far ahead the road is illuminated, so the sag equations use a 2 ft headlight height and a 1 degree upward beam divergence instead.
What is the minimum K value for a sag vertical curve?
It depends on design speed, because it is derived from that speed’s stopping sight distance: K = S² / (400 + 3.5S). At 50 mph, with a 425 ft design stopping sight distance, that gives 95.7, which TxDOT adopts as 96. Below roughly 55 mph a sag curve needs a larger K than a crest curve at the same speed, and above it the relationship reverses.
What is the difference between a crest and a sag vertical curve?
A crest curve turns the profile downward — the algebraic difference in grades A = G2 − G1 is negative. A sag curve turns it upward, with A positive. A road going from a +3% upgrade to a −2% downgrade is a crest; one going from −4% to +2% is a sag.
Does a larger K value mean a flatter curve?
Yes. K is length per percent of grade change, so a larger K spreads the same grade change over more distance. TxDOT flags K above 167 as a drainage threshold — not a maximum, but the point beyond which the profile is flat enough that a 0.30 percent grade is not reached within about 50 ft of the crest or sag, so pavement drainage needs closer attention.