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  2. Regular polygon - Wikipedia

    en.wikipedia.org/wiki/Regular_polygon

    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., they are concyclic points. That is, a regular polygon is a cyclic polygon.

  3. Apothem - Wikipedia

    en.wikipedia.org/wiki/Apothem

    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

  4. Constructible polygon - Wikipedia

    en.wikipedia.org/wiki/Constructible_polygon

    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 ...

  5. Polygon - Wikipedia

    en.wikipedia.org/wiki/Polygon

    Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, with the pentagram, a non-convex regular polygon (star polygon), appearing as early as the 7th century B.C. on a krater by Aristophanes, found at Caere and now in the Capitoline Museum. [40] [41]

  6. Pick's theorem - Wikipedia

    en.wikipedia.org/wiki/Pick's_theorem

    Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + ⁠ 96 / 2 ⁠ − 1 = 48 [1]. In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.

  7. Shoelace formula - Wikipedia

    en.wikipedia.org/wiki/Shoelace_formula

    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]

  8. 257-gon - Wikipedia

    en.wikipedia.org/wiki/257-gon

    The regular 257-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 257 is a Fermat prime , being of the form 2 2 n + 1 (in this case n = 3).

  9. Simple polygon - Wikipedia

    en.wikipedia.org/wiki/Simple_polygon

    The region inside the polygon (its interior) forms a bounded set [2] topologically equivalent to an open disk by the Jordan–Schönflies theorem, [10] with a finite but nonzero area. [11] The polygon itself is topologically equivalent to a circle, [12] and the region outside (the exterior) is an unbounded connected open set, with infinite area ...

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