Curve Radius Calculator

A simple horizontal curve is a circular arc joining two straight tangents, and its entire geometry follows from just the radius R and the deflection angle Delta between those tangents. Arc length is R times Delta in radians, tangent distance from the point of curvature to the point of intersection is R tan(Delta/2), external distance from the PI to the middle of the arc is R times (sec(Delta/2) minus 1), the middle ordinate is R times (1 minus cos(Delta/2)), and the long chord is 2R sin(Delta/2). Sharpness is often quoted instead as degree of curve, which under the arc definition is 5,729.578 divided by R in feet, that constant being 100 times 180 divided by pi. A 1 degree curve has a radius near 5,730 feet, a 10 degree curve about 573 feet. Safety is governed by the point-mass equation, minimum radius equals V squared divided by 15 times the sum of superelevation rate and side friction factor, with speed in mph and radius in feet. The 15 comes from gravity of 32.2 ft/s2 combined with the 1.467 conversion from mph to ft/s. Maximum superelevation ranges from about 4 percent in urban settings with frequent stops and icy conditions up to 10 or 12 percent on high-speed rural alignments, while the AASHTO side friction factor falls from roughly 0.17 at 20 mph to 0.08 at 80 mph, reflecting driver comfort rather than the friction the tire could actually supply. The minimum radius is therefore a comfort and steady-state criterion, not a skid threshold. In practice the controlling constraint is often something else entirely: sight distance blocked by a cut slope or barrier inside the curve, the length needed for superelevation runoff, or the spiral transition many state agencies require below a threshold radius.

The geometry of laying out circular curves entered civil practice through railway surveying in the 1830s and 1840s, when locating engineers had to stake long alignments across open country with a transit and a chain. The degree of curve convention, defined by the central angle subtending a 100 foot chord, exists precisely because a 100 foot chain was the field instrument: a chainman could set successive curve points by turning a constant deflection angle and pulling one chain length, without trigonometric tables or computation. North American railroads still use that chord definition, while highway practice adopted the arc definition that underlies the 5,729.578 constant. The transition spiral came next. The clothoid, whose curvature increases linearly with distance along the curve and therefore matches a driver or engineer turning at a constant rate, had been described mathematically by Leonhard Euler in the eighteenth century. It was reintroduced for railway use in Britain during the 1880s, and Arthur Newell Talbot's 1890 monograph The Railway Transition Spiral popularized it among American engineers. Superelevation and side friction were placed on a formal footing for motor traffic in the mid twentieth century by the American Association of State Highway Officials, renamed AASHTO in 1973, which consolidated its separate rural and urban policies into A Policy on Geometric Design of Highways and Streets in 1984. That volume, universally called the Green Book, is the source of the superelevation and friction tables and the minimum radius equation still used today, with NCHRP research through the 1990s refining how superelevation is distributed between the two.