Vertical Curve Calculator

How to use the vertical curve calculator

What this calculator does. This tool evaluates a symmetric parabolic vertical curve joining two roadway grades. It computes the elevation at a chosen horizontal distance, the elevations of the Point of Vertical Intersection (PVI) and End of Vertical Curve (EVC), the instantaneous grade, the parabolic offset from the initial tangent, the algebraic grade change, and the K value. It is useful for profile calculations and preliminary checking, but it does not determine whether a road design satisfies sight-distance, drainage, comfort, accessibility, or agency design criteria. The Federal Highway Administration explains that changes between straight roadway grades are normally joined with a parabolic vertical curve.

When to use it. Use the calculator when laying out a crest or sag curve, checking an elevation at a construction station, preparing a preliminary roadway profile, or verifying hand calculations before transferring values to CAD or survey software. It also helps students see how grade changes continuously along a parabola.

How to calculate. The calculator opens with a complete demonstration curve and an immediately available Excel workbook. (1) Choose Length unit; changing it converts all current length and elevation entries. (2) Replace Elevation of BVC, Initial gradient (g₁), Final gradient (g₂), and Length of curve with the project values. (3) Enter Horizontal distance from BVC (x) between zero and the curve length. Results, the profile chart, and the station table update live. (4) Read Elevation at selected distance as the main result, then check the PVI, EVC, grade, offset, curve type, grade change, and K value. (5) Select Download Excel to export the current canonical inputs and results as a validated OOXML workbook. Reset clears the demonstration data rather than restoring it; export is then disabled until all required fields are complete and valid again.

Input guide. Elevation of BVC is a required finite elevation in feet or meters; 1,000 ft is a realistic example. It shifts the entire profile vertically but does not change its shape. Do not confuse elevation with grade. Initial gradient (g₁) and Final gradient (g₂) are required signed percentages, such as +3 and – 3. Positive grade rises in the direction of increasing station, while negative grade falls; entering 0.03 when you mean 3% is a common hundredfold error. Length of curve is the required positive horizontal BVC-to-EVC distance; the example uses 300 ft. A longer length spreads the same grade change over more distance and increases K. Horizontal distance from BVC (x) is required and must lie from 0 through L; 120 ft is the startup value. It selects the point evaluated and highlighted but does not change the curve itself.

Output guide. Elevation at selected distance is the parabolic profile elevation at x. Elevation of PVI is where the incoming and outgoing tangent grades intersect for a symmetric curve; it is not generally on the parabolic curve. Elevation of EVC is the curve endpoint elevation. Instantaneous grade at x is the continuously changing slope at the selected station. Curve offset at x is curve elevation minus the initial tangent elevation and may be negative on a crest or positive on a sag. Curve type is crest when g₂ – g₁ is negative, sag when positive, and constant-grade when zero. Grade change is g₂ – g₁. K value is L divided by the absolute grade change in percentage points; a larger K means a flatter rate of grade change. The table reports station, curve elevation, initial-tangent elevation, offset, and grade. The chart plots those two elevation series and marks the selected station.

Worked example. With BVC elevation 1,000 ft, g₁ = +3%, g₂ = – 3%, L = 300 ft, and x = 120 ft, convert the grades to decimals: 0.03 and – 0.03. The elevation is 1,000 + 0.03 × 120 + ( – 0.03 – 0.03) × 120² ÷ (2 × 300) = 1,002.16 ft. The PVI elevation is 1,000 + 0.03 × 150 = 1,004.50 ft, while the EVC elevation is 1,000.00 ft. The grade at x is 0.03 + ( – 0.06 × 120 ÷ 300) = 0.60%. For deeper geometric context, see the FHWA Bridge Geometry Manual, which presents the general parabolic vertical-alignment equation.