Regular Polygon Miter and Layout

Saw miter and piece sizing to build any N-sided frame (octagon column wrap, hexagon planter, picture frame, segmented ring): each joint is mitered at 180/N degrees off square, the interior angle is (N-2) x 180/N, and the side relates to the across-flats width (s = flats x tan(180/N)) and across-corners diameter (s = corners x sin(180/N)), with perimeter and area (first-principles regular-polygon geometry; square 45, hexagon 30, octagon 22.5).

Regular N-gon frame: miter = 180/N deg off square at each end; interior angle = (N-2) x 180/N. Side from across-flats: s = flats x tan(180/N); from across-corners: s = corners x sin(180/N). Across-flats = s/tan(180/N), across-corners = s/sin(180/N), perimeter = N x s, area = (N x s^2)/(4 tan(180/N)).

Regular-polygon miter and layout geometry (the miter saw setting and the apothem/circumradius relations) - first-principles trigonometry; public domain.

Pure regular-polygon trigonometry, public; the number of sides and the target size are user-supplied.

Estimate. AHJ and licensed professional govern.

Field names used by the API: sides, size_mode, size_in, miter_angle_deg, interior_angle_deg, side_in, across_flats_in, across_corners_in, perimeter_in, area_in2

• Regular polygon the frame is a regular (equal-sided, equal-angled) N-gon built from straight pieces, mitered at each jointregular-polygon geometry