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Encyclopedia > Regular polygon
A regular pentagon

A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). Image File history File links Pentagon. ... Image File history File links Pentagon. ... A simple concave hexagon In geometry, two edges of a polygon may cross or even overlap in general. ... Look up polygon in Wiktionary, the free dictionary. ... In geometry, an equilateral polygon has all sides of the same length. ...

All regular polygons with the same number of sides are similar. // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ...

In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon. In geometry a digon is a polygon with two sides and two vertices. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For other uses, see Square. ... Look up pentagon in Wiktionary, the free dictionary. ... A regular hexagon. ... Look up Heptagon in Wiktionary, the free dictionary. ... For other uses, see Octagon (disambiguation). ... a regular decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all angles equal to 144Â°, therefore making each angle of a regular decagon be 144Â°. Its SchlÃ¤fli symbol is... In geometry, a dodecagon is a polygon with exactly twelve sides. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...

## Contents

Each angle of a regular n-gon has a measure of $(1-frac{2}{n})times 180$ (or equally of $(n-2)times frac{180}{n}$) degrees.

Alternately, the internal angle(s) of a regular n-gon is $frac{(n-2)pi}{n}$ radians ( or $frac{(n-2)}{2n}$ turns). ...

All vertices of a regular polygon lie on a common circle, i.e., they are concyclic points, i.e., every regular polygon has a circumscribed circle. In geometry, a set of points is said to be concyclic if they lie on a common circle. ... cle. ...

A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon. Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... In mathematics, any integer (whole number) is either even or odd. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ...

For n > 2 the number of diagonals is $frac{n (n-3)}{2}$, i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces. A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ...

## Area

Apothem of a hexagon

The area of a regular n-sided polygon is Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In geometry, the apothem of a regular polygon is a line drawn from the center of the polygon perpendicular to one of its sides. ... A regular hexagon. ...

$A=frac{nt^2}{4tan(pi/n)}$ where t is the length of a side. Also, the area is half the perimeter multiplied by the length of the apothem, a, (the line drawn from the centre of the polygon perpendicular to a side). That is A = a n t/2, as the length of the perimeter is n t. Or easier 1/2 p a.

For t=1 this gives The semiperimeter of a mathematical shape is defined as half of the shapes perimeter. ... In geometry, the apothem of a regular polygon is a line drawn from the center of the polygon perpendicular to one of its sides. ...

${frac{n}{4}} cot(pi/n)$

with the following values:

Sides Name Exact area Approximate area
3 equilateral triangle $frac{sqrt{3}}{4}$ 0.433
4 square 1 1.000
5 regular-pentagon $frac {1}{4} sqrt{25+10sqrt{5}}$ 1.720
6 regular-hexagon $frac{3 sqrt{3}}{2}$ 2.598
7 regular-heptagon   3.634
8 regular-octagon $2 + 2 sqrt{2}$ 4.828
9 regular-enneagon   6.182
10 regular-decagon $frac{5}{2} sqrt{5+2sqrt{5}}$ 7.694
11 regular-hendecagon   9.366
12 regular-dodecagon $6+3sqrt{3}$ 11.196
13 regular-triskaidecagon   13.186
20 regular-icosagon   31.569
100 regular-hectagon   795.513
1000 regular-chiliagon   79577.210
10000 regular-myriagon   7957746.893

The amounts that the areas are less than those of circles with the same perimeter, are (rounded) equal to 0.26, for n<8 a little more (the amounts decrease with increasing n to the limit π/12). A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... For other uses, see Square. ... Look up pentagon in Wiktionary, the free dictionary. ... A regular hexagon. ... Look up Heptagon in Wiktionary, the free dictionary. ... For other uses, see Octagon (disambiguation). ... A regular enneagon. ... a regular decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all angles equal to 144Â°, therefore making each angle of a regular decagon be 144Â°. Its SchlÃ¤fli symbol is... Categories: Math stubs | Polygons ... In geometry, a dodecagon is a polygon with exactly twelve sides. ... A regular triskaidecagon. ... In geometry, a pentadecagon is any 15-sided, 15-angled, polygon. ... HELLLOOOOO THERE ... Erchingers heptadecagon In geometry, a heptadecagon (or 17-gon) is a seventeen-sided polygon. ... An octadecagon is a polygon with 18 sides and 18 vertexes. ... An enneadecagon. ... A regular icosagon. ... A hectagon is a polygon with 100 edges. ... In geometry, a chiliagon (pronounced /ËˆkÉªli. ... This article or section does not cite its references or sources. ... The perimeter is the distance around a given two-dimensional object. ...

## Symmetry

The symmetry group of an n-sided regular polygon is dihedral group Dn (of order 2n): D2, D3, D4,... It consists of the rotations in Cn (there is rotational symmetry of order n), together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all axes pass through a vertex and the midpoint of the opposite side. The symmetry group of an object (e. ... This article may be confusing for some readers, and should be edited to enhance clarity. ... The smallest non-Abelian group has 6 elements. ... Some elementary examples of groups in mathematics are given on Group (mathematics). ... The triskelion appearing on the Isle of Man flag. ... Figures with the axes of symmetry drawn in. ...

## Nonconvex regular polygons

A pentagram

An extended category of regular polygons includes the star polygons, for example a pentagram, which has the same vertices as a pentagon, but connects alternating vertices. Image File history File links Pentagram_green. ... Image File history File links Pentagram_green. ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... A pentagram A pentagram (sometimes known as pentalpha or pentangle) is the shape of a five-pointed star drawn with five straight strokes. ... Look up pentagon in Wiktionary, the free dictionary. ...

Examples:

• Pentagram - {5/2}
• Heptagram - {7/2}, {7/3}
• Octagram - {8/3}
• Enneagram - {9/2}, {9/4}
• Decagram - {10/3}

A pentagram A pentagram (sometimes known as pentalpha or pentangle) is the shape of a five-pointed star drawn with five straight strokes. ... A heptagram or septegram is a seven-pointed star drawn with seven straight strokes. ... Enneagram Figure The Enneagram is a nine-pointed geometric figure. ...

## Polyhedra

A uniform polyhedron is a polyhedron with regular polygons as faces such that for every two vertices there is an isometry mapping one into the other. A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...

Plane tilings by regular polygons have been widely used since antiquity. ... A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ...

Results from FactBites:

 Polygon - LoveToKnow 1911 (1465 words) The term regular polygon is usually restricted to "convex" polygons; a special class of polygons (regular in the wider sense) has been named "star polygons" on account of their resemblance to star-rays; these are, however, concave. The Arabian geometers of the 9th century showed that the heptagon required the solution of a cubic equation, thus resembling the Pythagorean problems of "duplicating the cube" and "trisecting an angle." Edmund Halley gave solutions for the heptagon and nonagon by means of the parabola and circle, and by a parabola and hyperbola respectively. In general, the number of star polygons which can be drawn with the vertices of an n-point regular polygon is the number of numbers which are not factors of n and are less than In.
 PlanetMath: regular polygon (578 words) A regular polygon is a convex polygon such that all of its sides are congruent and all of its interior angles are congruent; that is, a polygon which is both equilateral and equiangular. The existence of many types of regular polygons is in contrast with the case for regular polyhedra, of which only five distinct types in Euclidean geometry. This is version 41 of regular polygon, born on 2002-02-19, modified 2007-07-16.
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