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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
Regular pentagon (n = 5) with side s, circumradius R and apothem a 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 area of a regular polygon is half its perimeter times the apothem (where the apothem is the distance from the center to the nearest point on any side).
Many specialized formulas apply to the areas of regular polygons. The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by =. This radius is also termed its apothem and is often represented as a.
The apothem is half the cotangent of /, and the area of each of the 14 small triangles is one-fourth of the apothem. The area of a regular heptagon inscribed in a circle of radius R is , while the area of the circle itself is ; thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.
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The area of a regular dodecagon of side length a is given by: = = (+) And in terms of the apothem r (see also inscribed figure), the area is: = = In terms of the circumradius R, the area is: [1]