How this calculator works
A radius drawn to the chord midpoint forms a right triangle with half the chord. Subtracting the midpoint distance from the radius gives sagitta. The calculation uses an equivalent rationalised expression to retain precision on shallow arcs. Only minor arcs, up to a semicircle, are modelled; this is circular geometry, not cable or beam deflection.
Formula
s = R - sqrt(R^2 - (c/2)^2) = (c/2)^2 / [R + sqrt(R^2 - (c/2)^2)]; R = c^2/(8s) + s/2; c = 2sqrt[s(2R - s)].
Worked example
A circle with 5 m radius and 6 m chord has a 4 m centre-to-chord distance, so sagitta is 1 m. The inverse expression returns radius 5 m from the same chord and sagitta.
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Frequently asked questions
Is curvature radius the same as focal length?
No. For a spherical mirror in the paraxial approximation, focal length is half the radius of curvature. The input here is the curvature radius.
Can the chord equal the diameter?
Yes. The arc is a semicircle and sagitta equals the radius. A longer chord is impossible.
Does this estimate a hanging wire sag?
No. A loaded cable follows a catenary or an appropriate load-dependent approximation. This page assumes a circular arc.