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Calculator · construction

Vertical Curve Elevation Calculator

Compute elevation at any station along a vertical curve using PVC elevation, entry/exit grades, curve length, and distance from PVC.

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Inputs

PVC Elevation (start of curve)

Entry Grade (G1)

Exit Grade (G2)

Length of Vertical Curve (L)

Distance from PVC (x)

Elevation at Point

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Get a plain-English breakdown of your result with practical next steps.

Elevation at Point—ft

The formula

How the
result is
computed.

How the Vertical Curve Elevation Formula Works

A vertical curve smoothly transitions between two road grades, ensuring driver comfort, adequate sight distance, and proper drainage on highways and local roads. The parabolic vertical curve is the universally accepted form in North American highway design, standardized by FHWA, AASHTO, and every state DOT.

The Standard Parabolic Elevation Equation

The elevation at any point along a vertical curve is computed with the parabolic equation:

y = y_PVC + (G₁ × x / 100) + ((G₂ − G₁) × x² / (200 × L))

Variable Definitions

y — Computed elevation (ft) at horizontal distance x from the PVC
y_PVC — Elevation at the Point of Vertical Curvature, where the curve leaves the back tangent
G₁ — Entry (back tangent) grade in percent; positive = uphill, negative = downhill
G₂ — Exit (forward tangent) grade in percent; positive = uphill, negative = downhill
L — Horizontal length of the curve in feet from PVC to PVT (Point of Vertical Tangency)
x — Horizontal distance in feet from the PVC to the point of interest (0 ≤ x ≤ L)

Why Engineers Use a Parabola

Highway engineers favor the parabola over circular arcs because it provides a constant rate of grade change per unit of horizontal distance. This uniformity simplifies construction staking, drainage design, and sight-distance verification, as detailed in the WSDOT Highway Surveying Manual, Chapter 11. The rate of grade change per foot equals (G₂ − G₁) / L, a constant throughout the entire curve length.

Crest vs. Sag Vertical Curves

Vertical curves divide into two types based on the algebraic difference in grades:

Crest curves — G₁ > G₂: the road transitions from a steeper uphill to a less steep uphill or downhill, forming a hilltop profile. Crest curves restrict stopping sight distance, so minimum length depends on AASHTO driver eye height (3.5 ft) and object height (2.0 ft) criteria.
Sag curves — G₁ < G₂: the road transitions from downhill to uphill or to a less steep downhill, forming a valley profile. Sag curve lengths are governed by headlight sight distance and driver comfort, with K values specified per design speed as outlined in the FHWA Speed Management Guide, Chapter 4.

Locating the High or Low Point

The elevation extreme (high point on a crest, low point on a sag) occurs where the instantaneous road grade equals zero. The horizontal distance from the PVC to this point is:

x_extreme = −G₁ × L / (G₂ − G₁)

Accurately locating the low point of a sag curve is critical for positioning storm drain inlets and preventing pavement flooding. The high point of a crest curve determines the sight-distance control station.

Worked Numerical Example

A highway engineer designs a sag vertical curve with these parameters: PVC elevation = 450.00 ft, G₁ = −3.5%, G₂ = +2.0%, L = 600 ft. Find the elevation at x = 300 ft:

y = 450.00 + (−3.5 × 300 / 100) + ((2.0 − (−3.5)) × 300² / (200 × 600))

y = 450.00 − 10.500 + (5.5 × 90,000 / 120,000)

y = 450.00 − 10.500 + 4.125 = 443.625 ft

The low point occurs at x = −(−3.5) × 600 / (2.0 − (−3.5)) = 2,100 / 5.5 ≈ 381.8 ft from the PVC, with a corresponding low-point elevation of approximately 442.98 ft.

AASHTO Minimum Length Standards

Minimum curve length uses the formula L = K × |G₂ − G₁|, where K is a design-speed-dependent rate-of-curvature constant. At 60 mph, AASHTO requires K ≥ 151 for crest curves and K ≥ 44 for sag curves. An absolute minimum equal to three times the design speed in feet (e.g., 180 ft at 60 mph) also applies regardless of the grade difference.

Practical Construction Applications

Setting finish-grade stakes for subbase and pavement lift elevations
Positioning storm drain inlets at sag curve low points
Calculating vertical clearances under bridge structures spanning sag curves
Verifying stopping and passing sight distances on crest curves
Generating earthwork cut-and-fill quantity estimates along the profile
Checking finished grades against design tolerances during construction inspection

Reference

Frequently asked questions

What is a vertical curve calculator and what does it compute?

A vertical curve calculator applies the standard parabolic highway formula to find the road elevation at any horizontal distance along a vertical curve. It requires five inputs: PVC elevation, entry grade G1, exit grade G2, curve length L, and distance x from the PVC. The result is the finished-road elevation at that station, used directly for construction staking, drainage design, and sight-distance verification on highway and roadway projects.

What is the difference between a crest and a sag vertical curve?

A crest vertical curve forms when the entry grade G1 exceeds the exit grade G2, creating a hilltop profile that can restrict stopping sight distance. A sag vertical curve forms when G1 is less than G2, creating a valley shape governed by headlight sight distance and driver comfort. Both curve types use the identical parabolic elevation formula, but their minimum length requirements differ substantially based on AASHTO design speed K-value tables for each type.

How do engineers determine the minimum length of a vertical curve?

Engineers apply the formula L = K x |G2 minus G1|, where K is a design-speed-dependent constant from AASHTO policy tables. At 60 mph, a crest curve requires K of at least 151 and a sag curve requires K of at least 44. An absolute minimum equal to three times the design speed in feet also applies. These minimums ensure adequate stopping sight distance on crest curves and sufficient headlight illumination distance on sag curves, consistent with FHWA and state DOT geometric design standards.

Where is the high or low point located on a vertical curve?

The high point on a crest curve or low point on a sag curve sits where the instantaneous road grade equals zero. The distance from the PVC to that point is x = negative G1 times L divided by (G2 minus G1). For example, with G1 = minus 3.5%, G2 = plus 2.0%, and L = 600 ft, the low point falls at 381.8 ft from the PVC. Locating the low point precisely determines where to place storm drain inlets to prevent pavement flooding.

What grade values are typical for highway vertical curve design?

Maximum grades for controlled-access highways generally range from 3% to 6% depending on design speed and terrain, while local roads may permit grades up to 10% to 12% on steep terrain. AASHTO recommends limiting grades to 3% to 4% on high-speed freeways operating at 70 mph or faster. The algebraic grade difference |G2 minus G1| directly drives minimum curve length requirements, meaning a transition from minus 4% to plus 4% (an 8% difference) demands a substantially longer curve than a 2% difference at the same design speed.

Does the vertical curve elevation formula work in metric units?

Yes. The parabolic formula y = y_PVC + G1 times x divided by 100 plus (G2 minus G1) times x squared divided by (200 times L) is dimensionally consistent and functions correctly in meters when L, x, and PVC elevation are all expressed in meters. Grades remain dimensionless percentages regardless of unit system. AASHTO and most U.S. DOTs publish companion metric design tables with K values expressed in meters per percent rather than feet per percent. Always confirm L and x share the same unit to avoid calculation errors.