This vertical curve length calculator helps highway and roadway designers size crest and sag vertical curves based on required sight distance. It follows standard geometric design equations from the AASHTO Green Book and is intended for preliminary design and education, not final contract documents.
A vertical curve provides a smooth transition between two different longitudinal grades on a roadway, improving ride comfort and ensuring that drivers can see far enough ahead to stop safely. A curve that is too short can:
The key design outputs are:
K = L / A, where A is the algebraic difference in grades. K is widely used in highway vertical curve design because it normalizes length by grade break.
A = |g₂ - g₁|.K = L / A, useful for comparing to design-speed guidelines.The main geometric parameter is the algebraic difference between the approach and departure grades, expressed as a percent:
A = |g₂ - g₁|
AASHTO provides slightly different forms of the sight-distance equations depending on whether the required sight distance S is shorter or longer than the vertical curve length L. The calculator checks both regimes and selects the appropriate formula automatically.
A representative crest vertical curve case with S ≤ L can be written using MathML as:
where:
For crest vertical curves, AASHTO typically assumes h₁ = 3.5 ft and h₂ = 2.0 ft for stopping sight distance. For sag vertical curves at night, the “object” is the illuminated point on the pavement from headlight beams, often represented with a headlight height of about 2.0 ft and a standard upward beam angle. The calculator uses standard values appropriate for highway vertical curve design.
When the required sight distance is longer than the curve length (S > L), alternative AASHTO equations apply. In that regime the relationship between L, S, and the height terms changes, and forms such as L = 2S - 200(h₁ + h₂)/A are used. The tool evaluates the correct form based on the inputs so that you do not need to select the case manually.
Once the minimum curve length L is computed, the K value is defined as:
K = L / A
K has units of ft/% and is widely used in design manuals because it provides a simple way to compare different combinations of grades and lengths. Larger K values correspond to “flatter” vertical curves that generally give better sight distance and ride comfort for a given design speed.
AASHTO and many highway agencies publish minimum recommended K values for a range of design speeds. The table below summarizes typical values for stopping-sight-distance-based design of two-lane and multilane rural highways; always check the latest edition of the AASHTO Green Book and your agency’s standards.
| Design Speed (mph) | Min Crest K | Min Sag K |
|---|---|---|
| 30 | 29 | 37 |
| 40 | 44 | 64 |
| 50 | 64 | 96 |
| 60 | 79 | 129 |
Compare the computed K from this calculator to guideline values:
Assume a two-lane rural highway crest vertical curve with:
Step 1 – Compute algebraic grade difference:
A = |g₂ - g₁| = |-2 - (+2)| = 4%
Step 2 – Use the appropriate AASHTO crest-curve equation for S ≤ L (the calculator checks the correct case). Using standard eye and object heights, the resulting minimum vertical curve length for this configuration is approximately:
L ≈ 171 ft
Step 3 – Compute K:
K = L / A ≈ 171 / 4 ≈ 43 ft/%
Step 4 – Compare to recommended K. For example, at a design speed of 50 mph, the table above shows Kmin, crest ≈ 64. Since 43 < 64, this crest vertical curve would not meet stopping sight distance requirements for 50 mph and should be lengthened or redesigned.
This calculator focuses on sight-distance-based minimum vertical curve length and K value for crest and sag curves. It does not substitute for a full geometric design or independent engineering judgment. Key assumptions and limitations include:
By pairing the curve length and K value from this calculator with design-speed-based guidelines and local standards, you can quickly screen alternative profiles and refine your vertical curve design before more detailed analysis.