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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.
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 .
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.
All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon . Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint.
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 ...
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 area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes; The volume of a sphere is 4 times that of a cone having a base of the same radius and height equal to this radius; The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter;