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How would I calculate the radius of a circular curve given only the tangent and arc lengths? An example being tangent = 260.28' and arc = 479.97'. The radius = 500.00'. I normally use Tangent = R tan(Δ/2) and Arc = (ΔπR)/180°. As you can see, both formulas use the central angle, or Δ, and the radius, or R. There's no geometric connection or relationship between the tangent length and arc length of any given circular curve. There must be an iterative approach for solving this. How would I know where to start an iteration, a best guess, for a problem like this? I assume from that point on an iteration would only approach the correct solution, not calculate an exact solution. After learning how to do this, I would like to apply it to a calculator program that I've written to solve circular curves. I only have a basic knowledge of calculus, but I've successfully passed precalculus algebra and analytic trigonometry in college.

Mark Hollinger
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  • We can convert minutes to radians to get arc length equal to $r\theta$. What quantity is the tangent of a circle? When I think of a tangent to a circle I think of a line that intersects it at one point. What does your formula represent? – John Douma Dec 18 '22 at 17:00
  • The quantity of the tangent is the distance from the point of tangency to the point of intersection. In surveying the back tangent is the distance from the point of curvature, or PC, to the point of intersection. The forward tangent is the distance from the point of intersection to the point of tangency, or PT. In practice the tangents are created first and the arc is fitted to the tangents. The tangent lines are perpendicular to the radial lines at the PC and PT. My formula represents the radial distance times the tangent of half of the delta, or the central angle between the radial lines. – Mark Hollinger Dec 20 '22 at 00:29

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