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Apothem of a hexagon Graphs of side, s; apothem, a; and area, A of regular polygons of n sides and circumradius 1, with the base, b of a rectangle with the same area. The green line shows the case n = 6. The apothem (sometimes abbreviated as apo [1]) of a regular polygon is a line
Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by
The radius of the inscribed circle is the apothem (the shortest distance from the center to the boundary of the regular polygon). For any regular polygon, the relations between the common edge length a, the radius r of the incircle, and the radius R of the circumcircle are:
The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals times the apothem (radius of the inscribed circle). All internal angles are 120 degrees . A regular hexagon has six rotational symmetries ( rotational symmetry of order six ) and six reflection symmetries ( six lines of symmetry ), making ...
The diagonals divide the polygon into 1, 4, 11, 24, ... pieces. [a] For a regular n-gon inscribed in a circle of radius , the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals n.
Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well.
The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensions are the 24-cell, snub 24-cell, 6-6 duoprism, 6-6 duopyramid. In 6 dimensions 6-cube, 6-orthoplex, 2 21, 1 22. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.
Let R be the radius of the arc which forms part of the perimeter of the segment, θ the central angle subtending the arc in radians, c the chord length, s the arc length, h the sagitta of the segment, d the apothem of the segment, and a the area of the segment.