Regular Polygon (Apothem, Across Flats/Corners, Area)

ScopeThe apothem, across-flats and across-corners, interior angle, and area for a hex or octagon. apothem = s/(2 tan(pi/n)), circumradius = s/(2 sin(pi/n)), area = n s^2/(4 tan(pi/n)), interior angle = (n-2)180/n. A 2 in hexagon measures 3.464 across the flats and 4.000 across the corners with a 10.39 area (interior 120 degrees); a 2 in octagon opens 135 degrees and 4.828 across the flats. The across-flats is the wrench size of a nut or the width of hex stock, the across-corners the diagonal, the apothem the inradius. For cutting hex/octagon stock, laying out a polygonal frame or tank, or checking a wrench. Irregular polygons and the miter/bolt pattern are separate. A shop and layout aid; verify critical dimensions on the work.

Formula or tableinterior angle = (n-2)180/n; apothem = s/(2 tan(pi/n)); circumradius = s/(2 sin(pi/n)); across-flats = 2 apothem; across-corners = 2 circumradius; area = n s^2/(4 tan(pi/n)).

Source and editionThe regular-polygon relations (apothem, circumradius, across-flats/corners, area) - standard plane geometry as in Machinery's Handbook (Industrial Press), by name; public domain.

Free accessPure plane geometry, public; the side count and side length are user-supplied.

What governsEstimate. AHJ and licensed professional govern.

API field namesnum_sides, side_length, across_flats, across_corners, area

• Apothem / circumradius a = s/(2 tan(pi/n)), R = s/(2 sin(pi/n)); across-flats = 2a, across-corners = 2Rplane geometry
• Area n s^2/(4 tan(pi/n)) = (1/2)(perimeter)(apothem)plane geometry
• Scope regular convex polygons; irregular shapes, miter, and bolt patterns are separatescope of this tile