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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
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.
A dodecagram is a 12-sided star polygon, represented by symbol {12/n}. There is one regular star polygon : {12/5}, using the same vertices, but connecting every fifth point. There are also three compounds: {12/2} is reduced to 2{6} as two hexagons , and {12/3} is reduced to 3{4} as three squares , {12/4} is reduced to 4{3} as four triangles ...
Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols. These properties apply to all regular polygons, whether convex or star: A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e
The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by A = 1 2 ⋅ p ⋅ r . {\displaystyle A={\tfrac {1}{2}}\cdot p\cdot r.} This radius is also termed its apothem and is often represented as a .
A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a cyclic polygon, and the circle is said to be its circumscribed circle or circumcircle. The inradius or filling radius of a given outer figure is the radius of the inscribed ...
The arc length, from the familiar geometry of a circle, is = The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of ):
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.