New York: Springer-Verlag, pp. Some regular polygons are easy to construct with compass and straightedge ; other regular polygons are not constructible at all. Beiler, A. Then the triangles and are similar. The most common example is the pentagramwhich has the same vertices as a pentagonbut connects alternating vertices. New York: Dover, p. That is, a regular polygon is a cyclic polygon. Regular polygons. New York: Wiley, pp. The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by.
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For a regular n -gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem  : p.
Examples include the Petrie polygonspolygonal paths of edges that divide a regular polytope into two halves, and seen as a regular polygon in orthogonal projection.
Beiler, A. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. Categories : Polygons. In the infinite limit regular skew polygons become skew apeirogons.
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Ship Orders Internationally · Home Services Handpicked Pros Happiness Guarantee · Amazon Inspire Digital Educational. In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral Regular polygons may be either convex or star. In the limit, a.
Hidden categories: Articles with short description. Guy gives additional information. Kern, W. The sides, and of such a triangle satisfy the Pythagorean theorem. In other projects Wikimedia Commons.
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Convexcyclicequilateralisogonalisotoxal. That is, a regular polygon is a cyclic polygon. All vertices of a regular polygon lie on a common circle the circumscribed circle ; i.
Drawthen since is similar toit follows that. Pythagorean Theorem 3D George Beck. A non-convex regular polygon is a regular star polygon.
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|New York: Wiley, p.
More generally regular skew polygons can be defined in n -space. This can be solved using elementary trigonometry as well as the explicit equations given above, and has solution.
A quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex. The inradius can be found by equating the area of the triangle with the sum of the areas of the three triangles, and having the inradii as altitudes, giving. Rothman, T. New York: Springer-Verlag, pp.