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Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, [11]: p. xi and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.
The only polygons that cannot be subdivided in this way are the special triangles considered above; therefore, only special triangles can appear in the resulting subdivision. Because each special triangle has area 1 2 {\displaystyle {\tfrac {1}{2}}} , a polygon of area A {\displaystyle A} will be subdivided into 2 A {\displaystyle 2A} special ...
A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if =, where r, s, k ≥ 0 and where the p i are distinct Pierpont primes greater than 3 (primes of the form +). [8]: Thm. 2 These polygons are exactly the regular polygons that can be constructed with Conic section, and the regular polygons ...
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
The area of a self-intersecting polygon can be defined in two different ways, giving different answers: Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region. For example, the central convex pentagon in the center of a ...
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
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). Fractals
The area of a regular 65537-gon is (with t = edge length) A = 65537 4 t 2 cot π 65537 {\displaystyle A={\frac {65537}{4}}t^{2}\cot {\frac {\pi }{65537}}} A whole regular 65537-gon is not visually discernible from a circle , and its perimeter differs from that of the circumscribed circle by about 15 parts per billion .