Arc Rise (Sagitta) from Radius and Chord
The inverse of the circular-arc tile: the rise (sagitta / middle ordinate) of an arc from a known radius and chord, rise = R - sqrt(R^2 - (chord/2)^2). A 24 in chord on a 20 in radius rises 4.0 in at midspan; also reports the arc length and central angle. Answers 'how high is the arc' when the radius is set (a curved wall, arch, or road curve). The chord cannot exceed the diameter. First-principles circle geometry.
Formula and source
Rise = R - sqrt(R^2 - (chord/2)^2), the inverse of R = (chord^2/4 + rise^2)/(2 rise); central angle = 2 x asin((chord/2)/R); arc length = R x angle. The chord cannot exceed the diameter (R >= chord/2).
Circular arc from a chord and radius (the sagitta / middle-ordinate relation) - first-principles circle geometry as in Machinery's Handbook (Industrial Press), by name; public domain.
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