Parabolic Segment Area and Arc Length

The area and curved length of a parabolic segment - a parabolic arch, a road or deck crown, a uniformly loaded cable sag, or a reflector cross-section. Area = (2/3) b h, exactly two-thirds of the b x h box (Archimedes), so it beats the triangle and loses to the rectangle; the exact arc length is (1/2)sqrt(b^2+16h^2) + (b^2/(8h)) ln((4h+sqrt(b^2+16h^2))/b). A 20 ft span, 5 ft rise arch encloses 66.7 area with a 22.96 ft curved edge; flatten the rise toward zero and the arc collapses to the 20 ft chord. To lay out the form, take off sheathing, or cut the rib. A circular segment and a true catenary are separate. A shop and layout aid; verify critical dimensions on the work.

area = (2/3) b h; root = sqrt(b^2 + 16 h^2); arc length = (1/2) root + (b^2/(8h)) ln((4h + root)/b). Rise->0 gives the chord.

The parabolic segment area (2/3) b h (Archimedes) and the exact parabolic arc length - standard geometry as in Machinery's Handbook (Industrial Press), by name; public domain.

Pure plane geometry, public; the span and rise are user-supplied measurements.

Estimate. AHJ and licensed professional govern.

Field names used by the API: base_span, rise_height, area, arc_length

• Area (2/3) b h, exactly two-thirds of the b x h box (Archimedes)plane geometry
• Arc length (1/2) sqrt(b^2+16h^2) + (b^2/(8h)) ln((4h+sqrt(b^2+16h^2))/b); -> chord as rise->0plane geometry
• Scope parabolic segment; circular segment and true catenary are separatescope of this tile