Vertical Curve Calculator With Stations and Elevations

A vertical curve calculator with stations uses the curve’s beginning station, length, entering and exiting grades, and starting elevation to find elevations and grades at stations along a parabolic road profile. Enter consistent stationing and units; the results describe the mathematical curve and should be checked against project plans and applicable design criteria.

Formula

The standard equal-tangent vertical curve is a parabola. At an offset from the curve’s beginning, its elevation and grade are:

• Elevation: E(x) = E₀ + (g₁/100)x + [(g₂ − g₁)/(200L)]x².
• Grade: g(x) = g₁ + [(g₂ − g₁)/L]x.
• Station: S(x) = S₀ + x.

Here, E(x) and E₀ are elevations in feet; x is the horizontal distance from the beginning of the curve in feet; L is the curve length in feet; g₁ and g₂ are entering and exiting grades in percent; and S₀ and S(x) are stations expressed as distances in feet. The grade formula returns percent. The elevation equation converts grade percentages to decimal slopes by dividing by 100.

For metric work, use metres for all distances and elevations, while retaining grades as percentages. Do not mix feet and metres in the same calculation. A station such as 10+00 is commonly a survey notation for 1,000 feet, not a value to enter as the text “10+00” in a calculator that expects a number.

Worked Example

Using the calculator defaults: beginning station S₀ = 1,000 ft, beginning elevation E₀ = 100 ft, length L = 400 ft, entering grade g₁ = 2%, exiting grade g₂ = −2%, and offset x = 200 ft.

1. Convert the entering grade to a decimal: 2/100 = 0.02. Its elevation contribution is 0.02 × 200 = 4 ft.
2. Calculate the curve term: [(-2 − 2)/(200 × 400)] × 200² = (-4/80,000) × 40,000 = −2 ft.
3. Add the contributions to the initial elevation: 100 + 4 − 2 = 102.000 ft.
4. Find the grade: 2 + [(-2 − 2)/400] × 200 = 2 − 2 = 0.000%.
5. Find the station: 1,000 + 200 = 1,200.00 ft, commonly written as station 12+00.

At the midpoint of this symmetrical crest curve, the grade is zero, while the elevation is 2 ft above the beginning elevation. The outputs are rounded to the precision specified in the calculator; the underlying equations use the entered values.

How Stations Work on a Vertical Curve

A station identifies a location along an alignment. In U.S. customary surveying, station 12+00 typically means 1,200 ft from the alignment’s station-zero reference, with the plus sign separating hundreds of feet. Some projects use different station intervals, metric chainage, or station equations, so verify the project’s convention rather than assuming notation.

The calculator uses an offset, x, measured forward from the vertical curve’s beginning (PC or BVC, depending on the plan’s terminology). If the beginning is station 10+00 and the target point is station 12+50, the offset is 250 ft. The curve’s end station is beginning station plus curve length, provided there is no project-specific station equation within that range.

For a metric alignment, chainage may be written in metres or kilometre-plus-metre notation. Convert the notation to a single numeric distance before calculating. A station label is not itself an elevation, and a vertical curve’s station reference does not imply that its elevation is zero.

When to Use the Calculator

This calculation is useful for checking profile elevations at regular intervals, estimating the grade at a point, or comparing a proposed profile with survey or layout data. It also helps explain how a vertical curve transitions smoothly between two tangent grades. It does not determine whether a design is suitable: sight distance, comfort, drainage, clearance, and project standards may govern the selected curve.

For any point on the curve, use an offset from its beginning. Confirm that the point lies within the curve: normally 0 ≤ x ≤ L. The calculator’s equations can return a mathematical value outside that interval, but that value is an extrapolation of the parabola, not necessarily the roadway profile beyond the curve.

Common Errors and Limits

• Entering a full station instead of an offset: If the curve begins at 10+00, do not enter 12+00 as x. Enter 200 ft for station 12+00.
• Mixing units: Curve length, offset, station distance, and elevation must use a consistent distance unit. Grades remain percentages in these equations.
• Forgetting the percent conversion: The elevation equation divides grades by 100. A 2% grade is 0.02 as a decimal slope.
• Reversing grade direction: Use the grade encountered entering the curve as g₁ and the grade leaving it as g₂. A downgrade is negative when the project’s direction of increasing station is forward and descending.
• Rounding too early: Keep full precision through intermediate calculations and round only the displayed result. Construction tolerances and required precision come from the project, not the calculator.

This model assumes a single parabolic vertical curve with constant rate of grade change and compatible stationing. It does not account for station equations, multiple connected curves, superelevation, cross slopes, or project-specific datum and adjustment conventions. Survey control, design documentation, and field procedures remain essential when elevations are used for engineering or construction.

Frequently Asked Questions

How do I calculate elevation at a station on a vertical curve?
Subtract the curve’s beginning station from the target station to get offset x. Substitute that offset, the beginning elevation, curve length, and both grades into the elevation equation. Check that the target is within the curve.

How do I convert station 10+50 to an offset?
First convert the station notation using the project’s convention. Under the common 100-ft station convention, 10+50 is 1,050 ft. If the curve begins at 10+00, the offset is 50 ft.

Can I use this calculator with metres?
Yes, if every distance and elevation is entered in metres. Keep grades in percent. Interpret station or chainage notation according to the project documents.

What does a zero grade at a station mean?
It means the curve’s tangent is horizontal at that point. For a simple crest curve changing from positive to negative grade, the zero-grade point is the high point; for a sag curve changing from negative to positive, it is the low point.

Is the curve’s high or low point always at its midpoint?
No. It is at the midpoint only when the entering and exiting grades have equal magnitudes and opposite signs. In general, set the grade equation to zero and solve x = −g₁L/(g₂ − g₁), then check whether that offset lies within the curve.

Does a vertical curve calculator establish that a road design is safe?
No. It computes profile values from supplied inputs. Design suitability depends on applicable criteria and other project conditions, so use the calculation as a check rather than a design approval.

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in sta | Beginning station | ft | 1000 | 0
in elev | Beginning elevation | ft | 100 | 0
in len | Curve length | ft | 400 | 0
in g1 | Entering grade | % | 2 | -
in g2 | Exiting grade | % | -2 | -
in x | Station offset from beginning | ft | 200 | 0
out elevation | Elevation at station | ft | elev + (g1 / 100) * x + ((g2 - g1) / (200 * len)) * x^2 | 3
out grade | Grade at station | % | g1 + ((g2 - g1) / len) * x | 3
out station | Station at point | ft | sta + x | 2