Vertical Curve Calculator

Q: How do you calculate the length of a vertical curve?

Multiply the K-value by the grade change: L = K × |A| where A = G₂ − G₁ . Example: a road goes from +3% to −2.5%. So A = −5.5% and |A| = 5.5. With K = 110 ft/%, the curve length is 110 × 5.5 = 605 ft . Most designers round that up to a clean number like 600 or 650 ft.

Q: What is a good K value for a vertical curve?

It depends on your design speed. Faster roads need bigger K values. Common AASHTO minimums (US, feet per percent) are: 30 mph: crest 19, sag 37 45 mph: crest 61, sag 79 55 mph: crest 114, sag 115 70 mph: crest 247, sag 181 These are minimums, not targets. Using a larger K gives a longer, flatter, smoother, safer curve.

Q: What is the minimum length of a vertical curve?

A common rule is L = 3V , where V is the design speed in mph. At 55 mph that is 165 ft. In metric it is L = 0.6V with V in km/h. This rule is about looks and comfort. It stops short, kinked-looking curves on flat grade changes. You must still check the sight distance K-value, which usually controls on faster roads.

Q: Why do engineers use a parabola instead of a circle for vertical curves?

A parabola changes grade at a steady rate from start to finish. That gives a smooth, even ride with no sudden jolt. It also makes the math simple. Elevations come from one easy equation, and the PVI always sits exactly halfway between the PVC and PVT. A circle would need harder math with almost no gain, since road grades are so flat that the two shapes look nearly the same.

Q: What does station 10+00 mean?

In US survey work, one station equals 100 feet. So 10+00 means 1,000 feet from the start of the project line. Station 13+50 is 1,350 feet. The number before the plus is the count of full 100-foot stations. The number after the plus is the extra feet. In metric work, one station equals 1,000 meters, so 1+000 means 1,000 m.

Q: How do you find the high point of a crest vertical curve?

Measure it from the PVC with: x = −G₁ × L ÷ A Example: G₁ = +3%, L = 600 ft, A = −5.5%. Then x = −(3)(600) ÷ (−5.5) = 327.3 ft past the PVC . Plug that x back into the curve equation to get the elevation. The same formula finds the low point of a sag curve.

Q: Why does the road not pass through the PVI?

The PVI is just where the two straight grade lines would cross if you extended them. The real pavement curves away from that corner. The gap between the PVI and the curve is: offset = |A| × L ÷ 800 For A = 5.5% and L = 600 ft, the curve sits 4.13 ft below the PVI on a crest, or above it on a sag.

Q: Why are sag curves designed for headlight sight distance?

On a sag curve at night, your headlights point at the dip in the road, so the beam only reaches so far. Daylight is not the problem; darkness is. The standard assumes headlights are 2.0 ft above the road and the beam aims 1 degree upward. That gives the sag formula L = A S² ÷ (400 + 3.5S) , where S is the stopping sight distance. Comfort and drainage are also checked, but headlights usually control the design.

Q: Where does the number 2158 come from in the crest curve formula?

It comes from the driver eye height and object height used by AASHTO. The eye is set at 3.5 ft and the object on the road at 2.0 ft. The constant is 200(√3.5 + √2.0)², which works out to about 2158 . That is why the crest formula is L = A S² ÷ 2158 when the curve is longer than the sight distance. In metric, with 1.08 m eye and 0.6 m object, the constant is 658.

Q: How does a downhill grade affect stopping sight distance?

It makes it longer. Gravity pulls the car forward, so brakes need more room to stop. The braking part of the formula is V² ÷ [30(a/32.2 + G)] , where G is the grade as a decimal. A downgrade is negative, which shrinks the bottom of the fraction and stretches the distance. At 55 mph, a 6% downgrade can add roughly 50 ft compared to flat ground.

Vertical Curve Calculator

Introduction

A vertical curve is the smooth dip or hump that joins two different road slopes. Without it, cars would hit a sharp bump where the grades meet. This Vertical Curve Calculator works out the shape of that curve using the same parabolic math that road designers use every day.

Type in your starting grade (G₁), your ending grade (G₂), and the curve length (L). The tool then gives you:

The grade change (A) and the K-value
Station and elevation at the PVC, PVI, and PVT
The high point of a crest curve or the low point of a sag curve
The elevation at any station you pick
The stopping sight distance (SSD) and the least curve length it needs

You also get a step-by-step solution, a station-elevation table you can print, and a chart of the curve. An AASHTO check tells you if your K-value passes for your design speed. Work in feet or meters, and use the sliders at the bottom to see how grade and length change the curve shape.

This calculator is built for civil engineers, surveyors, road designers, and students studying highway geometric design or getting ready for the FE and PE exams.

How to use our Vertical Curve Calculator

Enter your road grades, the curve length or K-value, and a known point on the curve. The calculator gives you the grade change (A), K-value, rate of grade change, the PVC, PVI, and PVT stations and elevations, the high or low point, an AASHTO check, a station-elevation table, and a curve diagram.

Unit System: Pick US Customary (feet) or Metric (meters). This only changes the labels. It does not convert your numbers.

Calculation Mode: Choose what you want to solve for: curve length, K-value, station elevation, required stopping sight distance, grade change, or the high/low point. Only the fields you need will show.

Initial Grade G₁ (%): Type the slope of the road going into the curve, in percent. Use a minus sign for a downhill grade, like −2.5.

Final Grade G₂ (%): Type the slope of the road leaving the curve, in percent. Use a minus sign for a downhill grade.

Curve Length L: Type the flat (horizontal) length from the PVC to the PVT in feet or meters. This is not the arc length.

K-Value: Type the rate of vertical curvature, which is the distance needed for each 1% of grade change. A bigger K makes a flatter, safer curve.

Design Speed: Type the road's design speed in mph or km/h. It is used to find the stopping sight distance and the AASHTO minimum K-value.

Stopping Sight Distance (SSD): Optional. Type your own SSD to override the math, or leave it blank so the tool figures it out from the design speed and grade.

Curve Type: Pick Crest (hill) or Sag (valley). If you enter both grades, the tool picks this for you.

Reference Point: Choose the point you know: the PVC (start of curve) or the PVI (where the two grades cross). The tool works out the other one.

PVC Station: Type the station at the start of the curve as a plain number. For STA 10+00, type 1000.

PVC Elevation: Type the profile elevation at the start of the curve.

PVI Station: Type the station where the two grade lines cross. It sits at the middle of the curve length.

PVI Elevation: Type the elevation at that crossing point. The curve itself does not touch this elevation.

Target Station: Type the station where you want the curve elevation. It should fall between the PVC and PVT.

Station Interval: Pick the spacing for the rows in the station-elevation table, or choose Custom and type your own. The PVC, PVI, PVT, and high/low point are always listed.

What Is a Vertical Curve?

A vertical curve is the smooth ramp that joins two different road slopes. Roads do not change grade all at once. If they did, cars would bounce, drivers would lose sight of the road ahead, and trucks could scrape the pavement. So engineers add a curved section of profile between the two slopes. In highway design, this curve is shaped like a parabola, which makes the grade change happen at a steady rate from start to finish.

Crest and Sag Curves

There are two kinds:

Crest curve (hill): the road goes from a steeper up-slope to a flatter or down-slope. The top of a hill is a crest. The big worry here is seeing far enough ahead, because the hill blocks your view.
Sag curve (valley): the road dips down and then rises. The big worry here is headlight range at night, plus rider comfort and water drainage.

Key Points on the Curve

PVC (Point of Vertical Curvature). Where the curve starts.
PVI (Point of Vertical Intersection). Where the two straight grade lines would cross. The road never actually touches this point.
PVT (Point of Vertical Tangency). Where the curve ends and the new grade begins.
High or low point. The spot where the road is flat (0% slope). This matters a lot for drainage, since water collects at low points.

Grades, A, K, and L

G₁ is the slope coming in and G₂ is the slope going out, both in percent. A downhill grade is negative. The algebraic grade difference is A = G₂ − G₁. If A is negative you have a crest; if A is positive you have a sag.

L is the length of the curve, measured flat along the station line from PVC to PVT, not along the pavement surface. The PVI always sits halfway between the PVC and PVT.

K is the rate of vertical curvature, found with K = L ÷ |A|. It tells you how many feet (or meters) of road are needed for each 1% of grade change. A bigger K means a longer, flatter, gentler curve. AASHTO design tables list minimum K values for each design speed, so a fast highway needs a much larger K than a slow local street.

Finding an Elevation on the Curve

Elevation at any distance x past the PVC comes from the parabola formula:

Elevation = Elev_PVC + (G₁ ÷ 100)·x + [A ÷ (200·L)]·x²

The first two parts give the straight tangent line. The last part is the tangent offset, or how far the real road sits below (crest) or above (sag) that straight line. Surveyors and grading crews use this to stake elevations every 25, 50, or 100 feet along the road.

Why Sight Distance Matters

Stopping sight distance (SSD) is how far ahead a driver must see to react and brake safely. It grows fast as speed goes up, and downhill grades make it longer because braking takes more room. On a crest curve, the hump itself limits how far you can see, so the curve must be long enough to give the driver that full distance. Designers check the curve length against the SSD rule and lengthen the curve if it comes up short. A curve that is too short is not just uncomfortable. It is unsafe.

Where Vertical Curves Are Used

Highways, city streets, driveways, parking lots, railroads, bike paths, airport runways, and pipelines all use vertical curves. Anywhere a grade changes and you need a smooth, safe, well-drained ride, a vertical curve is the tool engineers reach for.

Formulas used

Algebraic grade difference

A = G_2 - G_1

Rate of vertical curvature (K-value)

K = \frac{L}{|A|} \quad \Longleftrightarrow \quad L = K \cdot |A|

Parabolic curve elevation at distance x from PVC

Y = \text{Elev}_{PVC} + \frac{G_1}{100}x + \frac{A}{200L}x^2

PVC / PVT stations and elevations from the PVI

\text{Sta}_{PVC} = \text{Sta}_{PVI} - \frac{L}{2},\quad \text{Elev}_{PVC} = \text{Elev}_{PVI} - \frac{G_1}{100}\cdot\frac{L}{2},\quad \text{Elev}_{PVT} = \text{Elev}_{PVI} + \frac{G_2}{100}\cdot\frac{L}{2}

High/low (turning) point distance from PVC

x_{apex} = -\frac{G_1 L}{A}

Rate of grade change

r = \frac{A}{L}

Stopping sight distance (US customary / metric)

S = 1.47Vt + \frac{V^2}{30\left(\frac{a}{32.2}+G\right)} \qquad S = 0.278Vt + \frac{V^2}{254\left(\frac{a}{9.81}+G\right)}

Minimum curve length for sight distance (crest: C = 2158 or 658; sag: C = 400+3.5S or 120+3.5S)

L_{min} = \frac{|A|S^2}{C}\ \ (L > S) \qquad L_{min} = 2S - \frac{C}{|A|}\ \ (L < S)

Frequently asked questions

How do you calculate the length of a vertical curve?

Multiply the K-value by the grade change:

L = K × |A|

where A = G₂ − G₁.

Example: a road goes from +3% to −2.5%. So A = −5.5% and |A| = 5.5. With K = 110 ft/%, the curve length is 110 × 5.5 = 605 ft. Most designers round that up to a clean number like 600 or 650 ft.

What is a good K value for a vertical curve?

It depends on your design speed. Faster roads need bigger K values. Common AASHTO minimums (US, feet per percent) are:

30 mph: crest 19, sag 37
45 mph: crest 61, sag 79
55 mph: crest 114, sag 115
70 mph: crest 247, sag 181

These are minimums, not targets. Using a larger K gives a longer, flatter, smoother, safer curve.

What is the minimum length of a vertical curve?

A common rule is L = 3V, where V is the design speed in mph. At 55 mph that is 165 ft. In metric it is L = 0.6V with V in km/h.

This rule is about looks and comfort. It stops short, kinked-looking curves on flat grade changes. You must still check the sight distance K-value, which usually controls on faster roads.

Why do engineers use a parabola instead of a circle for vertical curves?

A parabola changes grade at a steady rate from start to finish. That gives a smooth, even ride with no sudden jolt.

It also makes the math simple. Elevations come from one easy equation, and the PVI always sits exactly halfway between the PVC and PVT. A circle would need harder math with almost no gain, since road grades are so flat that the two shapes look nearly the same.

What does station 10+00 mean?

In US survey work, one station equals 100 feet. So 10+00 means 1,000 feet from the start of the project line. Station 13+50 is 1,350 feet.

The number before the plus is the count of full 100-foot stations. The number after the plus is the extra feet.

In metric work, one station equals 1,000 meters, so 1+000 means 1,000 m.

How do you find the high point of a crest vertical curve?

Measure it from the PVC with:

x = −G₁ × L ÷ A

Example: G₁ = +3%, L = 600 ft, A = −5.5%. Then x = −(3)(600) ÷ (−5.5) = 327.3 ft past the PVC.

Plug that x back into the curve equation to get the elevation. The same formula finds the low point of a sag curve.

Why does the road not pass through the PVI?

The PVI is just where the two straight grade lines would cross if you extended them. The real pavement curves away from that corner.

The gap between the PVI and the curve is:

offset = |A| × L ÷ 800

For A = 5.5% and L = 600 ft, the curve sits 4.13 ft below the PVI on a crest, or above it on a sag.

Why are sag curves designed for headlight sight distance?

On a sag curve at night, your headlights point at the dip in the road, so the beam only reaches so far. Daylight is not the problem; darkness is.

The standard assumes headlights are 2.0 ft above the road and the beam aims 1 degree upward. That gives the sag formula L = A S² ÷ (400 + 3.5S), where S is the stopping sight distance.

Comfort and drainage are also checked, but headlights usually control the design.

Where does the number 2158 come from in the crest curve formula?

It comes from the driver eye height and object height used by AASHTO.

The eye is set at 3.5 ft and the object on the road at 2.0 ft. The constant is 200(√3.5 + √2.0)², which works out to about 2158.

That is why the crest formula is L = A S² ÷ 2158 when the curve is longer than the sight distance. In metric, with 1.08 m eye and 0.6 m object, the constant is 658.

How does a downhill grade affect stopping sight distance?

It makes it longer. Gravity pulls the car forward, so brakes need more room to stop.

The braking part of the formula is V² ÷ [30(a/32.2 + G)], where G is the grade as a decimal. A downgrade is negative, which shrinks the bottom of the fraction and stretches the distance.

At 55 mph, a 6% downgrade can add roughly 50 ft compared to flat ground.

What is the minimum grade needed for drainage on a vertical curve?

Near the high or low point the road goes nearly flat, and water can sit there. On curbed roads, designers keep at least a 0.3% grade within 50 feet of the level point.

That works out to a limit of about K = 167 ft/%. If your K is larger than that on a curbed street, add extra inlets or re-shape the gutter so water keeps moving.

Can both grades of a vertical curve be positive?

Yes. A curve from +1% to +4% is a real sag curve, and a curve from +4% to +1% is a real crest curve. The road keeps climbing the whole way.

In that case the high or low point falls outside the curve limits. That is normal and correct, not an error. It just means the pavement never reaches 0% slope between the PVC and PVT.

What is the rate of grade change (r) on a vertical curve?

It is how much the grade changes per unit of distance:

r = A ÷ L

For A = −5.5% over 600 ft, r = −0.00917% per foot, or about −0.92% per 100 ft.

Small r values mean a gentle curve. It is the same idea as K, just flipped: K = 1 ÷ |r|.

What is an unequal tangent vertical curve?

It is a curve where the PVI is not in the middle. One side is longer than the other.

It is really two parabolas joined at a shared point called the CVC. Each half is solved on its own using its own length and grades.

Designers use them when a fixed elevation, like a bridge deck, a driveway, or a utility line, forces the curve off center.

How do surveyors stake out a vertical curve in the field?

They pick a spacing, usually 25, 50, or 100 ft, and compute the curve elevation at each station with:

Elev = Elev_PVC + (G₁÷100)x + [A ÷ (200L)]x²

The PVC, PVI, PVT, and the high or low point are always included, even if they fall between the regular stations. Grade stakes are then set to those elevations so the crew can build to the design profile.

What happens if a vertical curve is too short?

Three problems show up. Drivers cannot see far enough over a crest to stop for something in the road. Riders feel a hard bump or a stomach drop. And on sag curves, headlights stop reaching far enough at night.

The fix is always the same: make the curve longer, which raises the K-value. Never shorten a curve just to save grading cost near a crest.

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