Vertical Curve Calculator for K-Value, PVC, PVT & Apex Data

Vertical Curve Calculator computes K-value, PVC, PVT, grade difference, and high or low point using K=L/A for equal-tangent parabolic crest and sag vertical curves.

Rate of Curvature (K-Value)

114.29 ft/%

Distance required for a 1% change in grade.

Absolute Grade Difference (A)

3.50%

Curve Type Crest Curve

Geometry Parabolic Profile

The absolute difference between entering and exiting grades.

Point of Vertical Curvature

800.00 ft

Point Type Curve Start (PVC)

Elevation 496.00 ft

The exact starting coordinate where the tangent meets the curve.

Point of Vertical Tangency

1200.00 ft

Point Type Curve End (PVT)

Elevation 497.00 ft

The exact ending coordinate where the curve transitions back to a straight tangent.

Apex / Turning Point

High Point

Linear Station 1028.57 ft

Elevation 498.29 ft

The highest or lowest physical elevation point occurring within the curve span.

Civil Engineering Note

Vertical curves are calculated as parabolas to provide a constant rate of change of grade. This creates a smooth and safe transition for vehicle dynamics and sight distance. K-Value determines the flatness of the curve.

Vertical Curve Calculator

This calculator solves the geometry of a parabolic vertical curve — the smooth transition between two road grades. Enter the curve length, entering and exiting grades, and the PVI station and elevation. The tool returns the K-value, absolute grade difference, PVC and PVT coordinates, and the location of any high or low point inside the curve.

Supports US Customary (feet) and Metric (metres). Suited for planning calculations, independent checks, and educational work on parabolic vertical curve geometry.

Final roadway, driveway, or site design must conform to the applicable local design standard. Where safety, drainage, or public-road approval is involved, results should be reviewed by a qualified professional engineer.

Vertical Curve Formulas

All calculations use the standard equal-tangent parabolic vertical curve centred on the PVI. Each block below corresponds to an output the calculator produces.

Grade Difference (A)

$$A = |g_1 - g_2|$$

A is the absolute difference between entering and exiting grades in percent. It represents the total grade change across the curve.

Rate of Curvature (K)

$$K = \frac{L}{A}$$

K is the horizontal distance (ft or m) required for each 1% change in grade. A larger K means a flatter curve. Design standards specify minimum K-values by design speed and curve type.

PVC Station

$$PVC_{station} = PVI_{station} - \frac{L}{2}$$

The curve starts at the PVC, half the curve length before the PVI along the horizontal alignment.

PVT Station

$$PVT_{station} = PVI_{station} + \frac{L}{2}$$

The curve ends at the PVT, half the curve length after the PVI.

PVC Elevation

$$PVC_{elev} = PVI_{elev} - \left(\frac{g_1}{100} \times \frac{L}{2}\right)$$

Project back from the PVI along the entering tangent grade to find the elevation at the curve start.

PVT Elevation

$$PVT_{elev} = PVI_{elev} + \left(\frac{g_2}{100} \times \frac{L}{2}\right)$$

Project forward from the PVI along the exiting tangent grade to find the elevation at the curve end.

Rate of Grade Change (r)

$$r = \frac{\dfrac{g_2}{100} - \dfrac{g_1}{100}}{L}$$

The constant rate at which grade changes per unit of horizontal distance along the parabola. Negative r indicates a crest; positive indicates a sag.

Apex Distance from PVC (x)

$$x = \frac{-\dfrac{g_1}{100}}{r}$$

Distance from PVC to the high or low point. The apex only exists inside the curve when $0 \le x \le L$. If x falls outside this range, no turning point lies within the curve span.

Apex Elevation

$$Elevation_x = PVC_{elev} + \left(\frac{g_1}{100} \times x\right) + \left(\frac{r}{2} \times x^2\right)$$

The parabolic elevation equation evaluated at distance x from the PVC. Gives the elevation at the high point (crest) or low point (sag).

A crest curve occurs when g₁ > g₂ — the road crests then falls. A sag curve occurs when g₁ < g₂ — the road dips then rises. The calculator identifies this automatically from your grade inputs.

What Each Input Means

System

Measurement System

US Customary (feet) or Metric (metres). All inputs and outputs use the selected unit. K-value is reported as ft/% or m/% accordingly.

Length of Curve (L)

Total horizontal length of the vertical curve from PVC to PVT. This is a horizontal distance, not a slope distance. For an equal-tangent curve, L is split equally on each side of the PVI.

g₁

Entering Grade (g₁)

Tangent grade approaching the PVI, in percent. Positive = upward slope; negative = downward slope in the direction of stationing. Example: +2.0% rises 2 ft per 100 ft of horizontal distance.

g₂

Exiting Grade (g₂)

Tangent grade departing the PVI, in percent. Same sign convention as g₁. The combination of g₁ and g₂ determines whether the result is a crest or sag curve and sets the total grade change A.

PVI Sta.

PVI Linear Station

The Point of Vertical Intersection (PVI) is where the two tangent grades meet. Enter its horizontal alignment station — the linear distance from the project datum or start of stationing. Used to compute PVC and PVT stations.

PVI Elev.

PVI Elevation

Elevation of the PVI — the theoretical intersection point of the two tangent lines. This point lies above (crest) or below (sag) the actual parabolic surface. Used with the grade inputs to back-calculate PVC and PVT elevations.

What Each Output Means

Rate of Curvature / K-Value

Horizontal distance (ft or m) needed per 1% change in grade. Higher K = flatter, more gradual curve. Compare your calculated K against the minimum K specified in your applicable design standard for the design speed and curve type.

Absolute Grade Difference (A)

Total change in grade across the curve in percent, calculated as |g₁ − g₂|. Reported alongside the curve type: Crest Curve when g₁ > g₂; Sag Curve when g₁ < g₂.

PVC

PVC Station and Elevation

The Point of Vertical Curvature — where the entering tangent ends and the parabola begins. Station is the horizontal distance along alignment; elevation is the profile grade elevation at that station.

PVT

PVT Station and Elevation

The Point of Vertical Tangency — where the parabola ends and the exiting tangent begins. Together with PVC, it defines the full extent of the vertical curve on the profile.

Apex

Apex / Turning Point (High or Low Point)

The point where the parabola's slope equals zero — the highest point on a crest curve or the lowest point on a sag curve. Reported only when it falls between PVC and PVT. If the computed x falls outside the curve span, the tool reports No Apex Present.

Worked Example — Crest Curve

A 400 ft crest curve with entering grade +2.0% and exiting grade −1.5%, centred on a PVI at station 1000 ft with elevation 500 ft.

Inputs

Length L 400 ft

Entering grade g₁ +2.0%

Exiting grade g₂ −1.5%

PVI station 1000.00 ft

PVI elevation 500.00 ft

Outputs

A 3.50%

K-value 114.29 ft/%

Curve type Crest

PVC station 800.00 ft

PVC elevation 496.00 ft

PVT station 1200.00 ft

PVT elevation 497.00 ft

High pt. station 1028.57 ft

High pt. elevation 498.29 ft

Step-by-step

Grade difference A

$$A = |2.0 - (-1.5)| = 3.50\%$$ Since g₁ > g₂, this is a crest curve.

K-value

$$K = \frac{400}{3.50} = 114.29 \text{ ft/\%}$$ Each percent of grade change occupies 114.29 ft of horizontal distance.

PVC station

$$PVC_{sta} = 1000 - 200 = 800.00 \text{ ft}$$

PVC elevation

$$PVC_{elev} = 500 - (0.02 \times 200) = 496.00 \text{ ft}$$

PVT station

$$PVT_{sta} = 1000 + 200 = 1200.00 \text{ ft}$$

PVT elevation

$$PVT_{elev} = 500 + (-0.015 \times 200) = 497.00 \text{ ft}$$

Rate of grade change (r)

$$r = \frac{-0.015 - 0.02}{400} = -0.0000875$$

Apex distance from PVC (x)

$$x = \frac{-0.02}{-0.0000875} = 228.57 \text{ ft from PVC}$$ Since 0 ≤ 228.57 ≤ 400, the apex is inside the curve.

High point station

$$sta = 800 + 228.57 = 1028.57 \text{ ft}$$

High point elevation

$$Elev = 496.00 + (0.02 \times 228.57) + \left(\frac{-0.0000875}{2} \times 228.57^2\right) = 498.29 \text{ ft}$$

Assumptions and Limitations

Understanding what this tool assumes helps you apply results correctly and identify cases where a more detailed analysis is needed.

Equal-tangent parabola only. The curve is divided equally on each side of the PVI (L/2 before, L/2 after). Unequal-tangent vertical curves require a separate method not covered here.
No sight-distance check. The calculator produces geometry only. It does not verify whether the K-value meets stopping sight distance (SSD) or passing sight distance (PSD) requirements for a given design speed. That check must be performed against the applicable design standard (e.g., AASHTO Green Book).
No drainage check. Sag curve low points determine where water collects. The tool locates the low point geometrically but does not assess inlet capacity, spread, or roadway drainage criteria.
Grades entered as percent. Enter grades in percent form (e.g., 2.0 for 2%, not 0.02). The tool handles the decimal conversion internally.
g₁ must differ from g₂. Identical grades produce A = 0 and an undefined K. No curve is needed in that case. The calculator flags this condition.
Apex outside the curve span. When computed x falls outside [0, L], the grade does not reverse within the curve. The tool reports this as No Apex Present.
Planning and educational use. Results are suitable for preliminary geometry, independent verification, and coursework. Final design for any public road, driveway, or graded site must follow the locally applicable design standard and be sealed by a licensed engineer where required by law.

References

American Association of State Highway and Transportation Officials (AASHTO). A Policy on Geometric Design of Highways and Streets ("Green Book"), current edition. Washington, DC: AASHTO. — Primary US reference for vertical curve K-value criteria by design speed and curve type.
Federal Highway Administration (FHWA). Geometric Design Publications. highways.dot.gov — FHWA resources on roadway geometric design including vertical alignment guidance.
Garber, N. J., & Hoel, L. A. Traffic and Highway Engineering, 5th ed. Cengage Learning. — Textbook derivation of the parabolic vertical curve formula, K-value, and sight-distance relationships.
Schoon, J. G. Geometric Design Projects for Highways: An Introduction, 3rd ed. ASCE Press. — Applied vertical curve calculations and worked examples including PVC/PVT and high/low point determination.
Wolf, P. R., Ghilani, C. D., & Ayers, P. R. Elementary Surveying: An Introduction to Geomatics, current edition. Pearson. — Parabolic vertical curve geometry and field-survey applications, including the rate-of-change formula used to locate turning points.
State DOT Roadway Design Manuals — individual state departments of transportation publish vertical curve design criteria including minimum K-values by facility type and design speed. Consult your state's current design manual for project-specific requirements.