Vertical Curve Calculator: Crest and Sag Curve Levels, High or Low Point and K Value

Levels along a parabolic crest or sag curve.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the two grades and the curve length.

Enter the chainage and level of the start of the curve.

Enter a chainage to find the level there.

Formulas used

  • RL = RL_{PVC} + \frac{g_1}{100}x + \frac{A}{200L}x^2
  • K = \frac{L}{|A|}
  • x_{hl} = -\frac{g_1L}{A}
  • e = \frac{AL}{800}

Worked example

With the default numbers in the calculator above:

  1. Algebraic difference in grade: -5 %
  2. K value (length for each percent of grade change): 40 m per percent
  3. Offset from the intersection of the grades to the curve: 1.25 m
  4. Reduced level at the chainage you chose: 51.8 m
  5. Grade of the road at that chainage: 0 %
  6. High point of the curve: distance from the start: 120 m (chainage 1120)
  7. High point of the curve: reduced level: 51.8 m

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is the K value?

The curve length divided by the change in grade in percent. The road design standard gives the minimum K for the sight distance and speed.

What is a crest and a sag?

A crest curve goes from an up grade to a down grade (it has a high point). A sag curve goes from a down grade to an up grade (it has a low point).

Where is the high or low point?

At a distance of minus the entering grade times the length, divided by the change in grade, from the start of the curve.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any road design or surveying text (vertical curves). Use the design standard for the K value and sight distance.

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