Calculate all five horizontal curve elements — tangent length (T), curve length (L), chord length (C), mid-ordinate (M), and external distance (E) — from curve radius and delta angle. Enter delta in degrees-minutes-seconds for precise survey work.
Stake horizontal curves with a Trimble S7 total station or Leica TS16 for sub-second angular precision.
Shop Express Tools →All horizontal curve elements are derived from two known values: radius (R) and delta angle (in radians). Delta must be converted from DMS to decimal degrees, then to radians before applying these formulas.
Road grading crews use horizontal curve geometry to set grade stakes along the curve centerline and offset lines. Before machine control was universal, crews staked curves by setting PC, PT, and then intermediate points using the deflection angle method — turning half the deflection angle for each station and measuring chord distance. Today, machine control GPS systems compute curve positions automatically, but field verification using curve elements remains essential for quality control.
Utility contractors working under road crossings or parallel to alignments use curve geometry to locate their work relative to the road centerline. When a utility line must follow a road curve, the offset distance changes continuously because the curve and the offset line have different radii. Understanding curve elements helps crews calculate accurate offsets at any station along the arc.
In some highway design traditions, curves are described by "degree of curve" (D) rather than radius. The arc definition (used by most state DOTs) defines D as the central angle subtended by a 100-foot arc: R = 5729.58 / D. A 1-degree curve has a radius of 5729.58 ft; a 10-degree curve has a radius of 572.96 ft. If your plans use degree of curve, convert to radius before using this calculator.
| Degree of Curve | Radius (ft) | Typical Use |
|---|---|---|
| 1° | 5729 ft | High-speed interstate (70+ mph) |
| 2° | 2865 ft | Primary highway (60 mph) |
| 4° | 1432 ft | Secondary highway (45–50 mph) |
| 6° | 955 ft | Rural road (40 mph) |
| 10° | 573 ft | Low-speed road or ramp (30 mph) |
| 20° | 286 ft | Tight ramp or parking (15–20 mph) |
Horizontal curve elements are the geometric properties that define a circular arc used to transition a road from one tangent direction to another. The five main elements are: T (tangent length — distance from the point of tangency to the PI), L (curve length — arc length along the road), C (chord length — straight line between PC and PT), M (mid-ordinate — perpendicular distance from chord midpoint to curve), and E (external distance — distance from PI to the midpoint of the curve). These elements are used by surveyors and engineers to stake out highway alignments.
The delta angle (also called the intersection angle or central angle) is the angle between the two tangent lines at the point of intersection (PI). It equals the central angle of the circular arc. Delta is expressed in degrees-minutes-seconds (DMS) for precise survey work. The same angle subtended at the center of the circle also defines the arc from PC (point of curvature) to PT (point of tangency).
Given radius (R) and delta angle (in radians), the formulas are: T = R × tan(Δ/2); L = R × Δ; C = 2R × sin(Δ/2); M = R × (1 − cos(Δ/2)); E = R × (sec(Δ/2) − 1). Delta must be converted from degrees-minutes-seconds to decimal degrees, then to radians (multiply by π/180) before using these formulas.
Curve length (L) is the actual arc distance measured along the road centerline from PC to PT — the distance a vehicle travels around the curve. Chord length (C) is the straight-line distance measured directly from PC to PT across the curve. For large-radius curves with small delta angles, the chord and arc are nearly equal. For tight curves with large delta angles, the difference becomes significant and both values are needed for staking.
The mid-ordinate (M) is the distance from the midpoint of the long chord to the midpoint of the arc. It is used to check sight distance on horizontal curves — AASHTO sight distance requirements often involve clearing obstructions (guardrails, cut slopes, vegetation) to at least the mid-ordinate distance inside the curve. A larger mid-ordinate means the curve is tighter and sight distance is more restricted.
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