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The apothem a can be used to find the area of any regular n-sided polygon of side length s according to the following formula, which also states that the area is equal to the apothem multiplied by half the perimeter since ns = p. = =.
For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem [4]: p. 72 (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n = 3 case. [5] [6]
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 7 R 2 2 sin 2 π 7 , {\displaystyle {\tfrac {7R^{2}}{2}}\sin {\tfrac {2\pi }{7}},} while the area of the circle itself is π R 2 ; {\displaystyle \pi R^{2 ...
where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula
Also, let each side of the polygon have length s; then the sum of the sides is ns, which is less than the circle circumference. The polygon area consists of n equal triangles with height h and base s, thus equals nhs/2. But since h < r and ns < c, the polygon area must be less than the triangle area, cr/2, a contradiction.
The span S of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) is ...
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A regular n-gon is constructible with straightedge and compass if and only if n = 2 k p 1 p 2...p t where k and t are non-negative integers, and the p i 's (when t > 0) are distinct Fermat primes. The five known Fermat primes are: F 0 = 3, F 1 = 5, F 2 = 17, F 3 = 257, and F 4 = 65537 (sequence A019434 in the OEIS).