Horizontal Curve Radius Calculator
The horizontal curve radius calculator computes all the geometric elements of a simple circular horizontal curve from the degree of curve (arc definition) and the intersection angle (delta). Enter the degree of curve (D) and intersection angle, and the calculator returns the radius, tangent length, arc length, long chord, and external distance. These are the standard design parameters used in US highway alignment calculations following AASHTO methods.
Horizontal curve formulas (arc definition)
R = 5,729.578 / D (feet)
T = R * tan(delta/2)
L = 100 * delta / D (or 2*pi*R*delta/360)
LC (long chord) = 2 * R * sin(delta/2)
E (external) = R * (1/cos(delta/2) - 1)
All angles in degrees converted to radians for trig. The constant 5,729.578 = 18,000 / pi.
Curve element uses
- Tangent length T: measured from the PC or PT to the PI along the tangents. Used to stake the curve beginning and end.
- Arc length L: used to compute stationing along the curve between PC and PT.
- Long chord LC: the straight-line distance from PC to PT. Used in staking by chord-offset methods.
- External E: the distance from the PI to the midpoint of the curve. Used in sight distance and clearing calculations.
Horizontal curve radius calculator: frequently asked questions
What is degree of curve in road design?
Degree of curve (D) is the central angle subtended by a 100-foot arc (arc definition, used in US highway design) or a 100-foot chord (chord definition, used in railroad design). A 1-degree curve is very gentle; a 10-degree curve is quite sharp.
What is the relationship between degree of curve and radius?
For the arc definition (highway): R = 5,729.58 / D, where R is the radius in feet and D is the degree of curve. For a 1-degree curve, R = 5,729.58 feet. For a 10-degree curve, R = 572.96 feet.
What are the elements of a simple horizontal curve?
The main curve elements are: R (radius), D (degree of curve), delta (intersection/deflection angle), T (tangent length from PC to PI), L (arc length), LC (long chord), and M (middle ordinate). All can be computed from any two of R and delta.
What is the PC and PT?
PC is the Point of Curvature (where the curve begins). PT is the Point of Tangency (where the curve ends). PI is the Point of Intersection of the two tangents. The curve provides a smooth transition between the two tangent alignments.
What minimum radius is required for highway design?
Minimum radius depends on design speed and superelevation (bank angle). AASHTO Green Book tables specify minimum radii: for 60 mph with 8% superelevation, minimum R = 1,000 feet. For 70 mph, minimum R = 1,500 feet. Local roads use smaller radii.
Official sources
- NOAA National Geodetic Survey: NGS surveying and geometry references.
- USGS: USGS geospatial and mapping resources.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.