Arc Rise (Sagitta) from Radius and Chord
The rise (sagitta / middle ordinate) of an arc from a known radius and chord.
Example
You enter
- Chord / span (in) 24
- Radius (in) 20
You get
- Rise (sagitta) at midspan 4 in
- Central angle 73.7398 deg
- Arc length 25.7400 in
Checked against published worked examples from: Mathwords (Bruce Simmons); Jerry Mahun, Open Access Surveying Library.
Details, formula, and sources
Scoperise = 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 or tableRise = 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).
Source and editionCircular 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.
Free accessPure circle geometry, public; the chord and radius are user-supplied.
What governsEstimate. AHJ and licensed professional govern.
API field nameschord_in, radius_in, rise_in, central_angle_deg, arc_length_in
- Chord and radius the chord (span) and the radius are user-supplied; rise = R - sqrt(R^2 - (chord/2)^2)circle geometry
- Minor arc returns the minor-arc rise; the chord must be no longer than the diametercircle geometry