Where two road grades meet, a parabolic vertical curve provides a smooth transition. Crest curves (g₁ > g₂) limit sight distance; sag curves (g₁ < g₂) affect headlight reach and drainage. Designers choose the curve length from the K value – length per percent change of grade – recommended for the design speed.
How to use the Vertical Curve Calculator
- Incoming grade g₁
- Outgoing grade g₂
- Curve length
- PVI station
- PVI elevation
- Station of interest
Results update as you type, and every field accepts the units shown in its menu.
How it works
Enter the incoming and outgoing grades in percent, the curve length, the station and elevation of the point of vertical intersection (PVI) and a station where you want the elevation.
Tips for an accurate estimate
- Measure angles with a digital angle finder or sliding bevel, not by eye.
- Make a test cut in scrap before cutting expensive stock.
- Small angle errors add up over many joints – check the first assembly before cutting the rest.
- Keep a written list of the dimensions and angles you set on the saw or machine.
Common mistakes to avoid
- Reading the saw scale as the angle from the edge rather than from square.
- Rounding intermediate results too early.
- Mixing degrees and percent grade.
Inputs explained
| Field | What to enter |
|---|---|
| Incoming grade g₁ | In % |
| Outgoing grade g₂ | In % |
| Curve length | Units: ft, m, in, yd, cm |
| PVI station | Units: ft, m, in, yd, cm |
| PVI elevation | Units: ft, m, in, yd, cm |
| Station of interest | Units: ft, m, in, yd, cm |
What each result means
- BVC station = [PVI station] – [Curve length]÷2
- BVC elevation = [PVI elevation] – [Incoming grade g₁]÷100×[Curve length]÷2
- EVC station = [PVI station] + [Curve length]÷2
- EVC elevation = [PVI elevation] + [Outgoing grade g₂]÷100×[Curve length]÷2
- High / low point elevation = [BVC elevation] + [Incoming grade g₁]÷100×([High ÷ low point station] – [BVC station]) + ([Outgoing grade g₂] – [Incoming grade g₁])÷100×([High ÷ low point station] – [BVC station])^2÷(2×[Curve length])
- K value (length per % grade change) = [Curve length]÷abs([Outgoing grade g₂] – [Incoming grade g₁])
Example values
How the main result changes when Incoming grade g₁ changes and everything else stays at the example values above:
| Incoming grade g₁ (%) | Elevation at station of interest (ft) |
|---|---|
| 1.5 | 497.333 |
| 2.25 | 496.333 |
| 3 | 495.333 |
| 3.75 | 494.333 |
| 4.5 | 493.333 |
| 6 | 491.333 |
Formula
E(x) = E_BVC + g₁ x + (g₂ − g₁) x² ÷ 2LBVC = PVI − L/2, E_BVC = E_PVI − g₁ L/2; EVC = PVI + L/2, E_EVC = E_PVI + g₂ L/2High/low point x = −g₁ L ÷ (g₂ − g₁); K = L ÷ |g₂ − g₁| (g in %)Worked example
Inputs — Incoming grade g₁: 3 %; Outgoing grade g₂: -2 %; Curve length: 600 ft; PVI station: 1,500 ft; PVI elevation: 500 ft; Station of interest: 1,400 ft.
Results — BVC station = 1,200 ft; BVC elevation = 491 ft; EVC station = 1,800 ft; EVC elevation = 494 ft; Elevation at station of interest = 495.333 ft; High / low point station = 1,560 ft; High / low point elevation = 496.4 ft; K value (length per % grade change) = 120 ft.
Key terms
- PVI
- Point of vertical intersection of the two tangent grades.
- BVC / EVC
- Beginning and end of the vertical curve.
Frequently asked questions
What is a vertical curve?
A parabola that joins two straight grades so the rate of change of grade is constant.
What is the K value?
The horizontal length needed for a 1% change in grade; AASHTO tables give minimum K for crest and sag curves by design speed.
Where is the high point of a crest curve?
At x = g₁L ÷ (g₁ − g₂) from the BVC, if it falls within the curve.
What formula does the Vertical Curve Calculator use?
E(x) = E_BVC + g₁ x + (g₂ − g₁) x² ÷ 2L.