Vertical Curve Calculator

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Walk-through

How to Use This Calculator

3 steps

1

Enter the entry and exit grades

Enter the entry grade (g1) and exit grade (g2) as signed percentages — positive for uphill, negative for downhill. These are the two straight tangent grades that the curve smoothly connects.

2

Set the curve length and PVI elevation

Enter the curve length (L) in feet — the horizontal distance from the start of curve (PVC) to the end of curve (PVT) — and the elevation at the point of vertical intersection (PVI), where the two tangent grades would meet if extended.

3

Enter a station and read the elevation

Enter the station you want to check, measured as a distance from the PVC (not an absolute highway station). The calculator instantly shows the elevation there, whether the curve is a crest or a sag, and the location of the curve's high or low point.

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Reference

Formula & Methodology

2 formulas

Parabolic vertical curve elevation

PVC_elev = PVI_elev − g1×L/200; elevation(x) = PVC_elev + g1×x/100 + (g2−g1)×x²/(200×L)

PVI is the tangent intersection at station L/2, not generally a point on the curve. PVC = PVI − g1×L/200. With grades in percent and x measured from PVC, E(x) = PVC + g1×x/100 + (g2−g1)×x²/(200×L).

High/low point location

x_hl = g1 × L / (g1 − g2)

The station (measured from the PVC) where the curve's slope is zero — the crest's high point or the sag's low point. Derived by differentiating the elevation equation and solving for zero slope. Undefined when g1 equals g2 (a straight tangent has no distinct turning point), and only physically meaningful when it falls between 0 and L.

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Glossary

Key Terms Explained

8 terms

Vertical curve A parabolic curve used in road and site design to smoothly transition between two straight grades, avoiding an abrupt change in slope that would jar vehicles or create a poor sight line.

PVI (Point of Vertical Intersection) The point where the two straight tangent grades would meet if extended. It sits at the midpoint of the curve length, station L/2 from the PVC, and is one of the standard inputs used to define the curve.

PVC (Point of Vertical Curvature) The start of the vertical curve, where the incoming tangent grade (g1) begins to curve. Station 0 in this calculator's convention.

PVT (Point of Vertical Tangency) The end of the vertical curve, where the curve straightens back out into the outgoing tangent grade (g2). Station L, the full curve length from the PVC.

Crest curve A vertical curve where the entry grade is greater than the exit grade (g1 > g2), so the road bends downward — like the top of a hill. Sight distance over a crest curve is a key design concern.

Sag curve A vertical curve where the entry grade is less than the exit grade (g1 < g2), so the road bends upward — like the bottom of a dip. Headlight sight distance at night is the key design concern for sag curves.

Grade The slope of a straight road or grading segment, expressed as a signed percentage: rise over run × 100. Positive is uphill in the direction of travel, negative is downhill.

Station A distance-based reference point along a route. This calculator measures the query station as a distance from the PVC (the start of the curve), in feet — not as an absolute highway station number.

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Scenarios

Real-World Examples

2 worked examples

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Road designer

Crest curve over a hill crown

Entry grade (g1) +2% Exit grade (g2) −1% Curve length (L) 400 ft PVI elevation 500 ft Station 150 ft

PVI is the tangent intersection at station L/2, not generally a point on the curve. PVC = PVI − g1×L/200. With grades in percent and x measured from PVC, E(x) = PVC + g1×x/100 + (g2−g1)×x²/(200×L). PVC 496 ft; PVT 498 ft; E(150) 498.15625 ft; vertex at 266.6667 ft: 498.6667 ft.

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Site grading contractor

Sag curve at the bottom of a driveway dip

Entry grade (g1) −2% Exit grade (g2) +3% Curve length (L) 300 ft PVI elevation 200 ft Station 150 ft

PVI is the tangent intersection at station L/2, not generally a point on the curve. PVC = PVI − g1×L/200. With grades in percent and x measured from PVC, E(x) = PVC + g1×x/100 + (g2−g1)×x²/(200×L). PVC 203 ft; PVT 204.5 ft; E(150) 201.875 ft; minimum at 120 ft: 201.8 ft.

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Reference

Cite This Calculator

APA & MLA

Use either format to cite this calculator in a paper, report, or resource list.

APA

MLA

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Deep Dive

Understanding Vertical Curves

3 chapters · 4 min read

A vertical curve is the parabolic transition road and site designers use to connect two straight grades without a sharp break in slope. This calculator applies the standard AASHTO equation used in highway and driveway design to find the elevation anywhere along a crest or sag curve, plus the high or low point that governs sight distance and drainage.

How the Vertical Curve Calculator works

PVI is the tangent intersection at station L/2, not generally a point on the curve. PVC = PVI − g1×L/200. With grades in percent and x measured from PVC, E(x) = PVC + g1×x/100 + (g2−g1)×x²/(200×L).

Inputs and what they mean

Entry grade (g1) and exit grade (g2) are the two straight tangent slopes the curve connects, entered as signed percentages. Curve length (L) is the horizontal distance from the PVC to the PVT, in feet — longer curves change grade more gradually, which is why higher-speed roads use longer curves for the same grade change. PVI elevation anchors the whole curve vertically; it's the elevation the two tangents would share if extended to meet at the curve's midpoint. Station is the point you want to evaluate, always measured as a distance from the PVC — entering an absolute highway station number instead of a PVC-relative distance is the single most common input mistake with this formula.

Limits and edge cases

When g1 equals g2 there's no curvature at all — it's a straight tangent, and there's no distinct high or low point to solve for, which the calculator flags directly. A station entered outside the 0–L range is mathematically extrapolated beyond the PVC or PVT and is not a valid point on the physical curve, so treat that result as informational only. This tool computes geometry only: it does not check minimum curve length for stopping sight distance, K-value design standards, or drainage adequacy at a sag's low point — those require checking against the AASHTO Green Book or your governing jurisdiction's design manual, and a low point without a drain inlet is a common real-world grading defect this calculator won't catch on its own.

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Questions

Frequently Asked Questions

6 questions

How do I tell if a vertical curve is a crest or a sag?+

Compare the entry grade (g1) to the exit grade (g2). If g1 is greater than g2, the road bends downward and it's a crest curve — think of the top of a hill. If g1 is less than g2, the road bends upward and it's a sag curve — think of the bottom of a dip. This calculator classifies the curve automatically from the grades you enter.

What exactly is the high/low point, and why does it matter?+

It's the station where the curve's slope equals zero — the literal top of a crest curve or the literal bottom of a sag curve. On a crest curve it's the critical point for stopping sight distance (drivers can't see past it). On a sag curve it's where surface water collects, so it's exactly where a drain inlet needs to go if the site isn't graded to shed water elsewhere.

What is the PVI, and why isn't the elevation I enter always the answer at my station?+

PVI is the tangent intersection at station L/2, not generally a point on the curve. PVC = PVI − g1×L/200. With grades in percent and x measured from PVC, E(x) = PVC + g1×x/100 + (g2−g1)×x²/(200×L).

What's the rate of grade change, and how does it relate to a 'K value'?+

Rate of grade change is (g2 − g1) / L — how many percentage points of grade change per foot of curve length. Road designers often use its reciprocal, K = L / (g2 − g1), as a design value: K tells you the curve length needed per percent of grade change to meet a minimum stopping-sight-distance standard. This calculator reports the rate of change directly; a smaller magnitude means a gentler, more gradual curve.

What is this calculator used for in practice?+

Road and driveway design (transitioning between grades smoothly and safely), site grading plans (setting finish elevations across a sloped lot), and drainage design (locating the low point of a sag curve where water will pool). It's the same parabolic-curve math used in civil engineering road profile sheets, scaled down to a quick lookup tool.

What units does this calculator use, and can I share a result?+

Grades are entered as percentages, length and station as feet, and elevation in whatever unit you're working in (typically feet) — the output elevation is returned in that same unit. All five inputs are encoded into the page URL, so copying the address bar link after you calculate is enough to share the exact scenario with a colleague.

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How to Use This Calculator

Enter the entry and exit grades

Set the curve length and PVI elevation

Enter a station and read the elevation

Formula & Methodology

Key Terms Explained

Real-World Examples

Road designer

Site grading contractor

Cite This Calculator

Understanding Vertical Curves

How the Vertical Curve Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Vertical Curve Calculator

Curve inputs

Elevation at your station

Elevation breakdown

How to Use This Calculator

Enter the entry and exit grades

Set the curve length and PVI elevation

Enter a station and read the elevation

Formula & Methodology

Key Terms Explained

Real-World Examples

Cite This Calculator

Understanding Vertical Curves

How the Vertical Curve Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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Guides & articles

Glossary terms