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Topic: Regular polygon

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Star polygon

Simple polygon

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	Polygon - LoveToKnow 1911
		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.
	www.1911encyclopedia.org /Polygon (1465 words)

	PlanetMath: regular polygon
		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.
	www.planetmath.org /encyclopedia/RegularPolygon.html (578 words)

	PlanetMath: regular polygon
		In ordinary usage, a regular polygon is a convex polygon with all its sides equal and all its angles equal, that is, a polygon that is both equilateral and equiangular.
		A regular triangle is also known as an equilateral triangle, and a regular quadrilateral is also known as a square.
		This is version 10 of regular polygon, born on 2002-02-19, modified 2006-11-03.
	planetmath.org /encyclopedia/RegularPolygon.html (338 words)

	Perimeter
		A polygon is 2-dimensional; however, perimeter is 1-dimensional and is measured in linear units.
		The perimeter of a regular hexagon is 18 centimeters.
		The perimeter of a regular pentagon is 100 centimeters.
	www.mathgoodies.com /lessons/vol1/perimeter.html (587 words)

	Polygon - MSN Encarta
		All polygons have an equal number of sides and vertices, and the sum of the interior angles of a polygon with n sides is 180° × (n – 2).
		If the sides of a polygon are of equal length and the angles are equal, the polygon is regular; otherwise it is irregular.
		The distance from the center of a regular polygon to a side is called its apothem.
	encarta.msn.com /encnet/refpages/RefArticle.aspx?refid=761553849 (279 words)

	Polygon Summary
		Regular polygons can be inscribed by a circle such that the circle is tangent to the sides at the centers, and circumscribed by a circle such that the sides form chords of the circle.
		Regular polygons are named to indicate the number of their sides or number of vertices present in the figure.
		If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.
	www.bookrags.com /Polygon (1346 words)

	polygons (Site not responding. Last check: 2007-11-05)
		The regular polygons are the analogues, in dimension two, of the regular polyhedra in dimension three and of the regular polytopes in dimension four.
		A regular polygon has all its sides equal and all its angles equal; its vertices are regularly distributed on a circle (their number n>2 is the order of the polygon).
		The faces of the uniform polyhedra, therefore of the regular polyhedra and of the Archimedes' polyhedra, are regular polygons.
	www.ac-noumea.nc /maths/amc/polyhedr/polygon_.htm (218 words)

	SparkNotes: Geometric Measurements: Area of Regular Polygons
		The radius of a regular polygon is a segment with one endpoint at the center and the other endpoint at one of the vertices.
		An apothem of a regular polygon is a segment with one endpoint at the center and the other endpoint at the midpoint of one of the sides.
		A central angle of a regular polygon is an angle whose vertex is the center and whose rays, or sides, contain the endpoints of a side of the regular polygon.
	www.sparknotes.com /math/geometry2/measurements/section6.rhtml (443 words)

	Figures and polygons
		A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.
		A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same.
		The sum of the angles of a parallelogram is 360 degrees.
	www.mathleague.com /help/geometry/polygons.htm (620 words)

	Math Forum: Ask Dr. Math FAQ: Regular Polygon Formulas
		It's a VERY famous theorem of Gauss that the only regular polygons with a prime number of sides that can be constructed with straightedge and compass are those for which the prime is one of the Fermat primes 3, 5, 17, 257, 65537,...
		The book also gives similar constructions for the regular polygons with 13 and 17 sides (for the regular 11-gon there's a construction using an angle-quinquesector, but it was too complicated for us to put into the book).
		Construction of a regular pentagon: Let N,S,E,W be the points of a circle C with center O in the four compass directions, and let M be the midpoint of ON.
	www.mathforum.org /dr.math/faq/formulas/faq.regpoly.html (1050 words)

	Regular Polygon Definition - Math Open Reference
		The vertex (plural: vertices) is a corner of the polygon.
		The apothem of a regular polygon is the line from the center to the midpoint of a side.
		The figure on the right is actually an example of an equilateral polygon since it has all sides the same length, but it is not a regular polygon because its interior angles are not all the same.
	www.mathopenref.com /polygonregular.html (390 words)

	All Elementary Mathematics - Study Guide - Geometry - Inscribed and circumscribed polygons. Regular polygons...
		A regular quadrangle is a square; a regular triangle is an equilateral triangle.
		A radius of a circumscribed circle is a radius of a regular polygon, a radius of a inscribed circle is its apothem.
		For the most of regular polygons it is impossible to express the relation between their sides and radii by an algebraic formula.
	www.bymath.com /studyguide/geo/sec/geo11.htm (538 words)

	Polygons
		Polygons are named according to the number of sides.
		A regular polygon's sides are all of the same length and its angles are the same size.
		polygon, triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, undecagon, dodecagon, concave polygon, convex polygon, regular polygon, irregular polygon
	www.mathsteacher.com.au /year7/ch09_polygons/05_polygon/pol.htm (225 words)

	Dummies::Sizing Up the Area of a Polygon
		Not only can polygons be classified by the number of sides they have and by their angles, but they can also be grouped according to some of their qualities.
		Theorem 5-11: The measure of a central angle in a regular polygon is equal to 360° divided by the number of sides of the polygon.
		Theorem 5-13: An apothem of a regular polygon is a perpendicular bisector to the side it's drawn to.
	www.dummies.com /WileyCDA/DummiesArticle/id-1195.html (802 words)

	polygon. The Columbia Encyclopedia, Sixth Edition. 2001-05
		In a regular polygon the sides are of equal length and meet at equal angles; all other polygons are not regular, although either their sides or their angles may be equal, as in the cases of the rhombus and the rectangle.
		The simplest regular polygons are the equilateral triangle, the square, the regular pentagon (of 5 sides), and the regular hexagon (of 6 sides).
		He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number (see number theory) of the form p = 2
	www.bartleby.com /65/po/polygon.html (275 words)

	Regular Polygons Project
		If we draw a circle around the polygon so that each vertex lies on the circle, then the center of the regular polygon will also be the center of the circle.
		For odd n, the median of a regular n-gon is the segment from a vertex to the center of the opposite side.
		Regular polygons with their diagonals drawn in can be viewed as graphs (with a vertex at each intersection), and it is helpful to reformulate our initial questions in terms of graph theory rather than simple plane geometry.
	www.math.rutgers.edu /~erowland/polygons-project.html (1433 words)

	Regular polyhedra
		Polygons are often identified with their interior regions.
		For a polygon it means that every closed curve in its interior can be continuously shrunk into a point while the deformation is being carried entirely inside the polygon.
		Regular faces cease to be regular polygons if of course they were regular to start with.
	www.cut-the-knot.org /do_you_know/polyhedra.shtml (1147 words)

	Poly Diameter, Radii, Pi
		Poly Diameter, Radii, Pi Regular Polygons are plane geometrical figures such as the square or octagon, having all inside corner angles equal, and all sides equal length.
		The only differences between even and odd sided regular polygons is: the even sided polygons have two (2) diameters that can be draw as straight lines end to end, and the odd sided polygons only have one (1) diameter that can be drawn as a straight line end to end.
		I believe the apothem is actually the true radius of all regular polygons, and twice the apothem is the true regular polygon diameter.
	www.theofficenet.com /~rad/Poly_Dia_Rad_PI.htm (916 words)

	Areas and Perimeters of Regular Polygons (Site not responding. Last check: 2007-11-05)
		These cases where the number of sides of the regular polygon is 3 or 4 are easy to calculate.
		To derive a formula for the area of a regular polygon if the number of sides is n requires applying some more trigonometry.
		Assume that point O is the center of the regular polygon and r, the distance from the center to a vertex, is called the radius of the polygon.
	www.algebralab.org /lessons/lesson.aspx?file=Geometry_AreaPerimeterRegularPolygons.xml (460 words)

	How Many Regular Polyhedrons Are There In This or Any Universe?
		Regular polygons are thus convex polygons whose vertex angles are all equal (or congruent) and whose sides are likewise all congruent.
		We will start with the simplest regular polygon, the equilateral triangle, manufacture all possible trihedral angles from just that unit, and then move up as far as needed to the point where the angle sum equals or exceeds 360 degrees.
		Three regular hexagons have a vertex angle sum of exactly 360 degrees, and they won't fold into a trihedral angle because there is no gap.
	www.iit.edu /~smile/ma8606.html (1460 words)

	polygonal labs » Calculating the Moment of Inertia of a non-regular convex Polygon
		First the polygon is triangulated by fanning around the centroid (center of mass), so that the centroid is part of every triangle.
		To calculate the ‘mass’ moment of inertia of a polygon you just have to multiply the area with a scalar that represents the density of the polygon, for example 2 [gramm per square meter].
		I.e imagine the points A and B in the polygon shown in the article to be missing; the polygon would be concave.
	lab.polygonal.de /2006/08/17/calculating-the-moment-of-inertia-of-a-convex-polygon (1720 words)

	Each Interior Angle
		polygon when all of its sides are of the same length and all of its angles are of the same measure.
		A regular polygon is both equilateral and equiangular.
		If a polygon is NOT REGULAR (such as the one seen at the right), you cannot use this formula.
	www.regentsprep.org /regents/math/geometry/GG3/LPoly2.htm (254 words)

	The Angles of a Regular Polygon
		Description: Give the program the number of sides of a regular polygon and it will give you the name of the polygon, the sum of the measure of the interior angles, the measure of each interior angle, and the measure of each exterior angle.
		He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number of the form p = 2
		The first five regular polygons with a prime number of sides that can be constructed using a straightedge and compass have 3, 5, 17, 257, and 65,537 sides.
	www.homeworkhotline.com /AnglesofaRegularPolygon.htm (264 words)

	Regular Polygons (Site not responding. Last check: 2007-11-05)
		If all the sides and all the angles of a polygon are equal the polygon is said to be regular.
		A regular polygon is equilateral and equiangular as is the quadrilateral shown here.
		Note that the angles and sides of each polygon are of equal measure.
	www.ul.ie /~cahird/polyhedronmode/regular.htm (97 words)

	Regular Polygon
		The center of a regular polygon is defined as the point inside the polygon which is equidistant to all the vertexes of it.
		The radii of a regular polygon are defined as the lines joining the vertexes and the center.
		Further evidence is the fact that the area of the infinite-sided regular polygon is in fact identical to the area of a circle.
	www.welltall.com /ymc/discovery/polygon.html (513 words)

	Totally Tessellated: Regular Tessellations, page 1/4
		We are now concerned with tessellations of regular polygons, since regular polygons are the simplest types of polygons.
		Next, it is easy to discover that squares (four-sided regular polygons) also tessellate, in a grid-like fashion.
		When trying to fit regular heptagons around a point or vertex, we see that the situation we had with regular pentagons happens again, overlap.
	library.thinkquest.org /16661/simple.of.regular.polygons/regular.1.html (204 words)

	Glossary
		The pentagram is a non-convex polygon; the Kepler-Poinsot solids are non-convex polyhedra.
		regular - A polygon is regular if its sides are equal and its angles are equal.
		Standardly, there are nine regular polyhedra: the five Platonic solids and the four Kepler-Poinsot solids, but others might be allowed, depending on the definition of polyhedron.
	www.georgehart.com /virtual-polyhedra/glossary.html (724 words)

	PlanetMath: regular polyhedron
		A regular polyhedron is a polyhedron such that
		There are only 5 regular polyhedra, as was first shown by Theaetetus, one of Plato's students.
		This is version 16 of regular polyhedron, born on 2002-02-19, modified 2006-09-28.
	planetmath.org /encyclopedia/Regular4.html (201 words)

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