Vertical Curve Length Calculator

Introduction: sizing a vertical curve from sight distance

A vertical curve is the parabola that carries a road from one grade to another. Its length is not chosen for comfort or aesthetics in the first instance; it is chosen so that a driver can see far enough ahead to stop. On a crest the curve itself blocks the view, and the length has to be long enough that the line of sight from a driver's eye to an object on the pavement clears the crest. On a sag the limit is at night: headlights point slightly upward, and if the curve is too sharp the beam strikes the pavement before the stopping distance is reached.

Those two situations produce different equations, and the difference is not cosmetic. The crest case is governed by two fixed heights — a 3.5 ft driver eye and a 2.0 ft object — and reduces to a single constant. The sag case is governed by a 2.0 ft headlight height and a 1 degree upward beam divergence, which introduces a term proportional to sight distance itself. A calculator that reuses the crest equation for sags with different heights, which is a common shortcut, gets the sag case badly wrong.

This page implements both AASHTO equations in both regimes, picks the geometrically consistent one, and reports the resulting rate of vertical curvature K so you can compare it against the published design controls. As a check on the implementation, feeding the AASHTO stopping sight distance for any design speed into these equations reproduces the Green Book K tables to the rounding: 425 ft at 50 mph gives a crest K of 83.7 against a published 84, and a sag K of 95.7 against a published 96.

How to use the vertical curve length calculator

1. Enter both grades with signs. Upgrade positive, downgrade negative, as a percent. A road rising at 2% into one falling at 2% is +2 and −2, giving an algebraic difference of 4%.
2. Pick crest or sag. The sign of the grade difference tells you which: negative is a crest, positive is a sag. The calculator flags a mismatch rather than silently computing the wrong case.
3. Set the sight distance. Choosing a design speed fills in the AASHTO stopping sight distance for that speed, or you can type your own — passing sight distance, decision sight distance, or an intersection-specific value.
4. Read K, not just L. K is length per percent of grade break, and it is what design manuals tabulate. Two projects with different grade breaks and the same design speed share a K, which is why agencies specify K rather than L.
5. Check the secondary criteria. A sag curve also has a rider-comfort criterion and, in kerbed sections, a drainage limit of K ≤ 167. Both are reported and either can govern over sight distance on a flat grade break. So can the design-speed minimum length of about three times the design speed, which is applied to crests as well.
6. Use the extra geometry. The results panel also gives the external distance E from the PVI down to the curve and the station of the high or low point measured from the PVC, plus the controlling length rounded up to the next 25 ft for plotting. Reset inputs puts the form back to the 50 mph crest it starts with.
7. Play the profile. The Sight Line game underneath draws the same geometry: it traces the driver's eye-to-object ray over a crest, or the headlight beam in a sag, and paints the pavement the driver cannot see in red as you stretch the curve.

The formulas, in both regimes

The algebraic difference in grades, in percent, is the driving parameter:

Crest vertical curves

With a driver eye height of 3.5 ft and an object height of 2.0 ft, the general sight-distance relation collapses to a constant:

so the two regimes are:

Sag vertical curves

At night the controlling geometry is a headlight 2.0 ft above the pavement whose beam diverges 1 degree upward from the vehicle's longitudinal axis. The 2.0 ft height contributes 200 × 2 = 400, and the 1 degree divergence contributes 200 × tan(1°) × S ≈ 3.5S, a term that grows with sight distance and has no counterpart in the crest case:

K, and the secondary criteria

The rate of vertical curvature normalises length by grade break, which is why it is what design manuals tabulate:

Note what that second form says: in the S ≤ L regime, K depends only on sight distance, not on the grade break at all. That is precisely why a single K covers every combination of A and L at one design speed, and it is the reason the Green Book can publish one number per speed.

For sags, two further criteria can govern. Rider comfort on a sag, where vertical acceleration is limited to about 1 ft/s², gives with V in mph; and on kerbed sections a drainage maximum of K ≤ 167 applies, so that a minimum 0.3% grade is maintained within 50 ft of the low point.

Reading the equations backwards

The same relations invert, which is what you want when a curve already exists and the question is whether it is long enough. For a crest the available sight distance is

and for a sag, where the S ≤ L branch comes from solving the quadratic in S,

The sag denominator 2A − 3.5 says something physical: below A = 1.75% the headlight beam's 1 degree upward divergence rises faster than the pavement does, the beam never comes back down to the road, and headlight sight distance stops being a constraint at all. WSDOT states exactly that limit in its sight distance chapter. The mini-game below uses these inverted forms to report the sight distance a curve actually delivers, and traces the ray or the beam to show you where it lands.

Where the curve sits, and where it turns over

Two further quantities come straight out of the parabola. The external distance, the vertical offset from the PVI down to the curve, is

and the high point on a crest or low point on a sag lies at a distance from the PVC of

which only falls inside the curve when the two grades have opposite signs. If both grades run the same way — a +4% into a +1%, for instance — the profile never turns over and the high point is simply the PVT. That matters for drainage: the turning point is where the profile grade goes flat, and it is the station a designer checks for ponding.

Plain-text formula: A = abs(g2 - g1); crest: L = A*S*S/2158 if S <= L else 2*S - 2158/A; sag: L = A*S*S/(400 + 3.5*S) if S <= L else 2*S - (400 + 3.5*S)/A; K = L/A; E = A*L/800; turning point x = L*g1/(g1 - g2), with S and L in feet, grades in percent.

Worked example: a 4% grade break at 50 mph

A road climbing at +2.0% meets one falling at −2.0%. The algebraic difference is A = 4.0%, and the negative sign of g₂ − g₁ confirms a crest. At a 50 mph design speed AASHTO gives a stopping sight distance of 425 ft.

1. Try the S ≤ L regime: L = 4.0 × 425² ÷ 2158 = 722,500 ÷ 2158 = 334.8 ft. Check the assumption: is S ≤ L? 425 is not ≤ 334.8, so this regime is inconsistent.
2. Try the S > L regime: L = 2 × 425 − 2158 ÷ 4.0 = 850 − 539.5 = 310.5 ft. Check: is S > L? 425 > 310.5, so this regime is the geometrically consistent one.
3. K from the consistent length is 310.5 ÷ 4.0 = 77.6. K from the S ≤ L form is 334.8 ÷ 4.0 = 83.7, which is the value AASHTO tabulates as 84.

Agency practice is normally to design to the tabulated K, that is to the S ≤ L form, because it is the conservative of the two and because it removes the regime question from the designer's hands. The calculator reports both so the difference is visible: on this curve the K-table length is 335 ft against a geometrically minimum 311 ft, a 24 ft margin.

For contrast, run the same 4% break as a sag: L = 4.0 × 425² ÷ (400 + 3.5 × 425) = 722,500 ÷ 1887.5 = 382.8 ft, giving K = 95.7 against the published 96. The comfort criterion gives L = 4.0 × 50² ÷ 46.5 = 215.1 ft, well below the sight-distance requirement, so sight distance governs; and K = 95.7 is comfortably under the 167 drainage limit.

Every published value is the computed value rounded up to the next whole number, with one exception worth knowing about: the 35 mph sag figure computes to 49.02 and is published as 49, rounded down rather than up. That single hundredth aside, the agreement across twenty-four values is the test that the constants 2158 and 400 + 3.5S are the right ones. An equation using a different denominator — the 1100 that appears when a 2.0 ft object height is wrongly substituted into the crest form, say — cannot reproduce the tables at any speed, let alone at all of them.

The last column is a separate control that has nothing to do with sight distance. Design manuals set a floor on vertical curve length of roughly three times the design speed in mph, so a 50 mph road gets at least 150 ft of curve however small the grade break. On a 0.6% break at 50 mph the sight-distance equation asks for about 50 ft; the 150 ft minimum is what actually goes on the plan. The calculator reports whichever of the three controls — sight distance, sag comfort, minimum length — is largest, and names it.

Notice also that the crest and sag columns cross over between 50 and 55 mph. Below that, sags demand the longer curve; above it, crests do. The reason is structural: the crest requirement grows as S² while the sag denominator grows with S, so the sag requirement is closer to linear in S at long sight distances.

Limitations and scope

This is preliminary design, not contract documents. The equations are the standard AASHTO forms, but a real profile has to satisfy drainage, superelevation transitions, structure clearances, utility conflicts, driveway grades, ADA cross-slope at crossings and the local agency's own supplements. Nothing here checks any of that.

US customary units only. The constants 2158 and 400 + 3.5S are specific to feet and percent. The metric forms use 658 and 120 + 3.5S with S and L in metres. Entering metres into this page produces nonsense.

Stopping sight distance is one objective among several. Passing sight distance produces far longer curves, and decision sight distance longer still. The design speed presets here fill in stopping sight distance; if you need another criterion, enter S directly.

The heights are fixed at the AASHTO passenger-car values. A 3.5 ft eye and 2.0 ft object suit a passenger car. Truck driver eye heights are considerably greater, which relaxes crest requirements, and the headlight geometry does not change for trucks at all, so a truck-controlled design is not simply a scaled version of this one.

Grades are assumed to be the tangent grades at the point of vertical intersection. Compound profiles, unsymmetrical vertical curves and curves overlapping horizontal curvature all need treatment this page does not attempt.

Rounding matters in practice. Curve lengths are normally rounded up to a convenient increment, commonly 50 or 100 ft, and stationing is set to whole or half stations. The unrounded figure here is a floor, not a length to put on a plan.

Common questions about vertical curve design

Sources checked for these formulas

Sources. The governing equations are the standard forms in AASHTO's A Policy on Geometric Design of Highways and Streets, the Green Book (AASHTO store listing). Because the Green Book itself is not free to read online, every constant and table entry on this page was checked against a state design manual that reproduces the same controls and is published in full: the WSDOT Design Manual, Chapter 1260 “Sight Distance” (PDF) and Chapter 1220 “Geometric Profile Elements” (PDF), both from WSDOT's manuals library. Specifically: Exhibit 1260-5 gives the crest forms L = 2S − 2158/A and S = (L + 2158/A)/2 for S > L, and L = AS²/2158 and S = √(2158L/A) for S < L, with the 3.5 ft eye and 2.0 ft object heights stated in section 1260.03; the constant is 100(√(2h₁) + √(2h₂))² = 100(√7 + 2)² = 2158. Exhibit 1260-7 gives the sag forms L = AS²/(400 + 3.5S) and L = 2S − (400 + 3.5S)/A together with their inverses, notes that the beam is measured to where a 1 degree upward divergence from a 2.0 ft headlight meets the roadway, and records that values of A below 1.75% sit inside that divergence and need no sag sight-distance check. Exhibit 1260-1 supplies the design stopping sight distances, the crest K_c and sag K_s design values and the minimum curve length VCL_m reproduced in the table above, all twenty-four K values of which were recomputed from the equations and agree. Chapter 1220 states the sag comfort criterion L = AV²/46.5. The drainage maximum of K = 167 maintains a minimum 0.3% grade within 50 ft of the turning point on a kerbed section. All constants are US customary. This page is for preliminary design and education; final roadway design must follow the current Green Book edition and the governing agency's own standards.