Circular Arc Layout
Radius, arc length, and central angle of a circular arc from a measured chord (span) and rise (sagitta / middle ordinate) at midspan: R = (chord^2/4 + rise^2) / (2 x rise), central angle = 2 x acos((R - rise)/R), arc length = R x angle - the everyday layout question for an arch, curved trim, sheet-metal radius, or road curve (first-principles circle geometry).
Formula and source
Radius R = (chord^2 / 4 + rise^2) / (2 x rise); central angle = 2 x acos((R - rise) / R); arc length = R x angle. The rise is the sagitta (middle ordinate) at midspan.
Circular arc from a chord and rise (the sagitta / middle-ordinate relation) - first-principles circle geometry as in Machinery's Handbook (Industrial Press), by name; public domain.
Audience
This tile is built for carpentry and the adjacent professions in the Cross-Trade Utilities group. The interactive calculator runs entirely in your browser. No account, no fee, no advertising, no tracking.
Related tiles
Posture
Rough Logic answers the math question the working professional asks on the job. The site is a calm, fast, ad-free, account-free, ever-free reference. It does not interpret code. It does not replace the licensed professional. It does not store your inputs. The Authority Having Jurisdiction governs all installations and inspections.