Vertical Curve Calculator
Free Vertical curve Calculator for civil engineering projects. Enter dimensions to get material lists and cost estimates.
Formula
y = PVC Elev + G1(x) + (r/2)(x^2)
The elevation at any point on a vertical curve equals the PVC elevation plus the entering grade times the distance from PVC, plus half the rate of grade change times the distance squared. The rate of grade change r = (G2 - G1) / L, where G1 and G2 are the entering and exiting grades and L is the curve length.
Worked Examples
Example 1: Crest Vertical Curve
Problem: Design a 600 ft crest vertical curve connecting a +3% grade to a -2% grade. PVC at Station 100+00, Elevation 500.00 ft.
Solution: A = |-2 - 3| = 5%\nr = (-0.02 - 0.03)/600 = -0.0000833/ft\nPVI: Sta 103+00, Elev 509.00\nPVT: Sta 106+00, Elev 512.50\nHigh point: x = -0.03/(-0.0000833) = 360 ft\nHigh point Sta: 103+60, Elev 505.40
Result: High point at Sta 103+60, Elev 505.40 ft
Example 2: Sag Vertical Curve
Problem: Calculate key points for a 400 ft sag curve from -4% to +2%, PVC at Sta 50+00, Elev 250.00.
Solution: A = |2 - (-4)| = 6%\nr = (0.02 - (-0.04))/400 = 0.00015/ft\nLow point: x = 0.04/0.00015 = 266.7 ft\nLow point Sta: 52+66.7\nLow point Elev: 250 + (-0.04)(266.7) + 0.5(0.00015)(266.7^2) = 244.67
Result: Low point at Sta 52+66.7, Elev 244.67 ft
Frequently Asked Questions
What is a vertical curve in road design?
A vertical curve is a parabolic transition between two tangent grades (slopes) in the vertical profile of a roadway. Vertical curves provide a smooth transition for vehicles traveling over hills (crest curves) or through valleys (sag curves). They are essential for driver safety because they ensure adequate sight distance, provide comfortable ride quality, and allow proper drainage. The curve is defined by its length and the algebraic difference between the entering and exiting grades.
What is the difference between crest and sag vertical curves?
A crest vertical curve occurs when the road goes over a hill, meaning the entering grade is higher than the exiting grade (the algebraic difference is positive when going from uphill to downhill). A sag vertical curve occurs in a valley where the road changes from downhill to uphill. Crest curves are primarily designed for stopping sight distance, as the hilltop can obstruct the driver view. Sag curves are designed for headlight sight distance at night and driver comfort, as the gravitational change at the bottom of a sag creates an uncomfortable sensation.
How do I calculate the minimum vertical curve length?
Minimum vertical curve length depends on the design speed, the algebraic difference of grades (A), and the required sight distance. For crest curves, the minimum length L = A x S^2 / 2158 (when S is less than L), where S is the stopping sight distance and A is the algebraic grade difference in percent. For sag curves, L = A x S^2 / (200 x (3.5 + 0.035S)). These formulas assume specific driver eye height and object height values from AASHTO standards. The calculated length should be rounded up to the nearest 100-foot station increment.
How do I find the high or low point on a vertical curve?
The high point (on crest curves) or low point (on sag curves) occurs where the instantaneous grade equals zero. The distance from the PVC (point of vertical curvature) to the high or low point is x = -G1 / r, where G1 is the entering grade in decimal form and r is the rate of grade change per station (r = (G2 - G1) / L). The elevation at that point is calculated by substituting x back into the curve equation. This point is critical for drainage design, as it determines where water will collect on the roadway surface.
Can I use Vertical Curve Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.