query: what is the name of polygon with 7 sides

pos: Heptagon regular skew polygon around the perimeter is called the petrie polygon. Heptagon In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of "", a Latin-derived numerical prefix, rather than "", a Greek-derived numerical prefix) together with the Greek suffix "-agon" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is {7}. The area ("A") of a regular heptagon of side length "a" is given by: This can be seen by

pos: Heptagon regular skew polygon around the perimeter is called the petrie polygon. Heptagon In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of "", a Latin-derived numerical prefix, rather than "", a Greek-derived numerical prefix) together with the Greek suffix "-agon" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is {7}. The area ("A") of a regular heptagon of side length "a" is given by: This can be seen by

pos: Constructible polygon Constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known. Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of

pos: Reuleaux triangle the United Kingdom has made 20-pence and 50-pence coins in the shape of a Reuleaux heptagon. The Canadian Loonie dollar coin uses another Reuleaux polygon with 11 sides. Similar methods can be used to enclose an arbitrary simple polygon within a curve of constant width, whose width equals the diameter of the given polygon. The resulting shape consists of circular arcs (at most as many as sides of the polygon), can be constructed algorithmically in linear time, and can be drawn with compass and straightedge. Although the regular-polygon based Reuleaux polygons all have an odd number of circular-arc sides, it

pos: Heptagram Heptagram A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name "heptagram" combines a numeral prefix, "hepta-", with the Greek suffix "-gram". The "-gram" suffix derives from "γραμμῆ" ("grammē") meaning a line. In general, a heptagram is any self-intersecting heptagon (7-sided polygon). There are two "regular" heptagrams, labeled as {7/2} and {7/3}, with the second number representing the vertex interval step from a regular heptagon, {7/1}. This is the smallest star polygon that can be drawn in two forms, as irreducible fractions. The two heptagrams are sometimes called the "heptagram" (for {7/2}) and

pos: Heptagon Heptagon In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of "", a Latin-derived numerical prefix, rather than "", a Greek-derived numerical prefix) together with the Greek suffix "-agon" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is {7}. The area ("A") of a regular heptagon of side length "a" is given by: This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices

neg: Triacontadigon zonogon (a 2"m"-gon whose opposite sides are parallel and of equal length) can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular triacontadigon", "m"=16, and it can be divided into 120: 8 squares and 7 sets of 16 rhombs. This decomposition is based on a Petrie polygon projection of a 16-cube. A triacontadigram is a 32-sided star polygon. There are seven regular forms given by Schläfli symbols {32/3}, {32/5}, {32/7}, {32/9}, {32/11}, {32/13}, and {32/15}, and eight compound star figures with

neg: Concyclic points of a regular polygon of any number of sides are concyclic. A tangential polygon is one having an inscribed circle tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle. A convex quadrilateral is orthodiagonal (has perpendicular diagonals) if and only if the midpoints of the sides and the feet of the four altitudes are eight concyclic points, on what is called the eight-point circle. Concyclic points In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are the same

neg: 56 (number) of an angle trisector is allowed since 56 = 2 × 7. 56 (number) 56 (fifty-six) is the natural number following 55 and preceding 57. 56 is: The maximum determinant in an 8 by 8 matrix of zeroes and ones is 56. Plutarch states that the Pythagoreans associated a polygon of 56 sides with Typhon and that they associated certain polygons of smaller numbers of sides with other figures in Greek mythology. While it is impossible to construct a perfect regular 56-sided polygon using a compass and straightedge, a close approximation has recently been discovered which it is claimed might

neg: Polygon lengths of the sides of a polygon do not in general determine the area. However, if the polygon is cyclic the sides "do" determine the area. Of all "n"-gons with given sides, the one with the largest area is cyclic. Of all "n"-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic). The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer: The idea of a polygon has been generalized in various ways. Some of the more important include: The word "polygon" comes from

neg: Bicentric polygon Bicentric polygon In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides. Every triangle is bicentric. In a triangle, the radii "r" and "R" of the incircle and circumcircle respectively are related by the equation where "x" is the

neg: Triacontagon parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular triacontagon", "m"=15, it can be divided into 105: 7 sets of 15 rhombs. This decomposition is based on a Petrie polygon projection of a 15-cube. A triacontagram is a 30-sided star polygon. There are 3 regular forms given by Schläfli symbols {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same vertex configuration. There are also isogonal triacontagrams constructed as deeper truncations of the regular pentadecagon {15} and pentadecagram {15/7}, and inverted pentadecagrams {15/11},

neg: Tangential polygon Tangential polygon In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an "incircle"). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices. All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi and kites. A convex polygon has an incircle if and

neg: Parallelogon parallelogon is a zonogon, but hexagonal parallelogons enable the possibility of nonconvex polygons. A parallelogram can tile the plane as a distorted square tiling while a hexagonal parallelogon can tile the plane as a distorted regular hexagonal tiling. Parallelogon A parallelogon is a polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides. A parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides; a

neg: Parallelogon Parallelogon A parallelogon is a polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides. A parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides; a four-sided parallelogon is a parallelogram. In general a parallelogon has 180-degree rotational symmetry around its center. Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms. In general they all have central inversion symmetry, order 2. Every convex

neg: Rectilinear polygon to an internal node. The simplest rectilinear polygon is an axis-aligned rectangle - a rectangle with 2 sides parallel to the x axis and 2 sides parallel to the y axis. See also: Minimum bounding rectangle. A golygon is a rectilinear polygon whose side lengths in sequence are consecutive integers. A rectilinear polygon which is not a rectangle is never convex, but it can be orthogonally convex. See Orthogonally convex rectilinear polygon . A monotone rectilinear polygon is a monotone polygon which is also rectilinear. A T-square is a fractal generated from a sequence of rectilinear polyons with interesting properties.

neg: Regular polygon 2π radians or one full turn. As the number of sides, n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a myriagon) the internal angle is 179.964°. As the number of sides increase, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line. For this reason, a circle is not a polygon with

neg: Corresponding sides and corresponding angles example if in a polygon with the side sequence "abcde" and another with the corresponding side sequence "vwxyz" we have vertex angle "A" appearing between sides "a" and "b" then its corresponding vertex angle "V" must appear between sides "v" and "w". Corresponding sides and corresponding angles In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has

neg: Tetradecagon For the "regular tetradecagon", "m"=7, and it can be divided into 21: 3 sets of 7 rhombs. This decomposition is based on a Petrie polygon projection of a 7-cube, with 21 of 672 faces. The list defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection. The regular tetradecagon is used as the shape of some commemorative gold and silver Malaysian coins, the number of sides representing the 14 states of the Malaysian Federation. A tetradecagram is a 14-sided star polygon, represented by symbol {14/n}. There are two regular star polygons: {14/3} and

neg: Circumscribed circle (adding up to 180° or π radians). For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal, and sides 2, 4, 6, ... are equal). A cyclic pentagon with rational sides and area is known as a Robbins pentagon; in all known cases, its diagonals also have rational lengths. In any cyclic "n"-gon with even "n", the

neg: Icosioctagon of each other and have half the symmetry order of the regular icosioctagon. Coxeter states that every zonogon (a 2"m"-gon whose opposite sides are parallel and of equal length) can be dissected into "m"("m" − 1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular icosioctagon", "m" = 14, and it can be divided into 91: 7 squares and 6 sets of 14 rhombs. This decomposition is based on a Petrie polygon projection of a 14-cube. An icosioctagram is a 28-sided star polygon. There are

neg: Equilateral polygon exists a principal diagonal "d" such that and a principal diagonal "d" such that Triambi are equilateral hexagons with trigonal symmetry: Equilateral polygon In geometry, three or more than three straight lines (or segment of a line) make a polygon and an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, it need not be equiangular (need not have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon need not be a convex polygon: it

neg: Tetracontaoctagon sides are parallel and of equal length) can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular tetracontaoctagon", "m"=24, and it can be divided into 276: 12 squares and 11 sets of 24 rhombs. This decomposition is based on a Petrie polygon projection of a 24-cube. A tetracontaoctagram is a 48-sided star polygon. There are seven regular forms given by Schläfli symbols {48/5}, {48/7}, {48/11}, {48/13}, {48/17}, {48/19}, and {48/23}, as well as 16 compound star figures with the same vertex

neg: Rectilinear polygon Rectilinear polygon A rectilinear polygon is a polygon all of whose edge intersections are at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons. In many cases another definition is preferable: a rectilinear polygon is a polygon with sides parallel to the axes of Cartesian coordinates. The distinction becomes crucial when spoken about sets of polygons: the latter definition would imply that sides of all polygons in the set are aligned with the same coordinate axes. Within the framework of the second definition it is natural

neg: Equilateral polygon with 60° internal angles. An equilateral quadrilateral is called a rhombus, an isotoxal polygon described by an angle α. It includes the square as a special case. A convex equilateral pentagon can be described by two angles α and β, which together determine the other angles. Concave equilateral pentagons exist, as do concave equilateral polygons with any larger number of sides. An equilateral polygon which is cyclic (its vertices are on a circle) is a regular polygon (a polygon that is both equilateral and equiangular). A tangential polygon (one that has an incircle tangent to all its sides) is equilateral

neg: Skew polygon petrie polygon has. This is 5 sides for a 5-cell, 8 sides for a tesseract and 16-cell, 12 sides for a 24-cell, and 30 sides for a 120-cell and 600-cell. When orthogonally projected onto the Coxeter plane these regular skew polygons appear as regular polygon envelopes in the plane. The "n"-"n" duoprism and dual duopyramids also have 2"n"-gonal petrie polygons. (The tesseract is a 4-4 duoprism, and the 16-cell is a 4-4 duopyramid.) Skew polygon In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The "interior" surface

neg: Icosihexagon and have half the symmetry order of the regular icosihexagon. Coxeter states that every zonogon (a 2"m"-gon whose opposite sides are parallel and of equal length) can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular icosihexagon", "m"=13, and it can be divided into 78: 6 sets of 13 rhombs. This decomposition is based on a Petrie polygon projection of a 13-cube. An icosihexagram is a 26-sided star polygon. There are 5 regular forms given by Schläfli symbols: {26/3}, {26/5}, {26/7},

neg: Cyclogon a polygon rolls over a straight line. Let it be assumed that the regular polygon rolls over the edge of another polygon. Let it also be assumed that the tracing point is not a point on the boundary of the polygon but possibly a point within the polygon or outside the polygon but lying in the plane of the polygon. In this more general situation, let a curve be traced by a point z on a regular polygonal disk with n sides rolling around another regular polygonal disk with m sides. The edges of the two regular polygons are assumed

neg: Polygon Polygon In elementary geometry, a polygon () is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or "circuit". These segments are called its "edges" or "sides", and the points where two edges meet are the polygon's "vertices" (singular: vertex) or "corners". The interior of the polygon is sometimes called its "body". An "n"-gon is a polygon with "n" sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic

neg: Dual polygon Dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals. In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely. For example, the dual of a highly acute isosceles triangle is an obtuse

neg: Hectogon whose opposite sides are parallel and of equal length) can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular hectogon", "m"=50, it can be divided into 1225: 25 squares and 24 sets of 50 rhombs. This decomposition is based on a Petrie polygon projection of a 50-cube. A hectogram is a 100-sided star polygon. There are 19 regular forms given by Schläfli symbols {100/3}, {100/7}, {100/9}, {100/11}, {100/13}, {100/17}, {100/19}, {100/21}, {100/23}, {100/27}, {100/29}, {100/31}, {100/33}, {100/37}, {100/39}, {100/41}, {100/43}, {100/47},

neg: 360-gon (a 2"m"-gon whose opposite sides are parallel and of equal length) can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular 360-gon", "m"=180, and it can be divided into 16110: 90 squares and 89 sets of 180 rhombs. This decomposition is based on a Petrie polygon projection of a "180-cube". A 360-gram is a 360-sided star polygon. There are 47 regular forms given by Schläfli symbols {360/7}, {360/11}, {360/13}, {360/17}, {360/19}, {360/23}, {360/29}, {360/31}, {360/37}, {360/41}, {360/43}, {360/47}, {360/49}, {360/53}, {360/59},

neg: Enneacontahexagon states that every zonogon (a 2"m"-gon whose opposite sides are parallel and of equal length) can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular enneacontahexagon", "m"=48, and it can be divided into 1128: 24 squares and 23 sets of 48 rhombs. This decomposition is based on a Petrie polygon projection of a 48-cube. An enneacontahexagram is a 96-sided star polygon. There are 15 regular forms given by Schläfli symbols {96/5}, {96/7}, {96/11}, {96/13}, {96/17}, {96/19}, {96/23}, {96/25}, {96/29}, {96/31}, {96/35},

neg: 120-gon opposite sides are parallel and of equal length) can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular 120-gon", "m"=60, and it can be divided into 1770: 30 squares and 29 sets of 60 rhombs. This decomposition is based on a Petrie polygon projection of a "60-cube". A 120-gram is a 120-sided star polygon. There are 15 regular forms given by Schläfli symbols {120/7}, {120/11}, {120/13}, {120/17}, {120/19}, {120/23}, {120/29}, {120/31}, {120/37}, {120/41}, {120/43}, {120/47}, {120/49}, {120/53}, and {120/59}, as well

neg: Concave polygon a convex polygon. As with any simple polygon, the sum of the internal angles of a concave polygon is ("n" − 2) radians, equivalently 180°×("n" − 2), where "n" is the number of sides. It is always possible to partition a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polygons as possible is described by . A triangle can never be concave, but there exist concave polygons with "n" sides for any "n" > 3. An example of a concave quadrilateral is the dart. At least one interior angle

neg: Icositetragon dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular icositetragon", "m"=12, and it can be divided into 66: 6 squares and 5 sets of 12 rhombs. This decomposition is based on a Petrie polygon projection of a 12-cube. <BR>A regular triangle, octagon, and icositetragon can completely fill a plane vertex. An icositetragram is a 24-sided star polygon. There are 3 regular forms given by Schläfli symbols: {24/5}, {24/7}, and {24/11}. There are also 7 regular star figures using the same vertex arrangement:

neg: Hexacontagon can be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular hexacontagon", "m"=30, and it can be divided into 435: 15 squares and 14 sets of 30 rhombs. This decomposition is based on a Petrie polygon projection of a 30-cube. A hexacontagram is a 60-sided star polygon. There are 7 regular forms given by Schläfli symbols {60/7}, {60/11}, {60/13}, {60/17}, {60/19}, {60/23}, and {60/29}, as well as 22 compound star figures with the same vertex configuration. Hexacontagon In geometry, a hexacontagon or

neg: Simple polygon Simple polygon In geometry a simple polygon is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pair-wise to form a closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition then being understood to define a polygon in general. The definition given above ensures the following properties: Two edges meeting at a corner are usually required to form an angle that is not straight (180°); otherwise, the collinear line segments will be considered parts of a single side. Mathematicians typically use "polygon"

neg: Cupola (geometry) axis, and the mirror planes pass through the "z"-axis and bisect the sides of the base. They also either bisect the sides or the angles of the top polygon, or both. (If "n" is even, half of the mirror planes bisect the sides of the top polygon and half bisect the angles, while if "n" is odd, each mirror plane bisects one side and one angle of the top polygon.) The vertices of the base can be designated V through V, while the vertices of the top polygon can be designated V through V. With these conventions, the coordinates of

neg: Equilateral polygon Equilateral polygon In geometry, three or more than three straight lines (or segment of a line) make a polygon and an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, it need not be equiangular (need not have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon need not be a convex polygon: it could be concave or even self-intersecting. All regular polygons and isotoxal polygons are equilateral. An equilateral triangle is a regular triangle

neg: Isothetic polygon Isothetic polygon An isothetic polygon is a polygon whose alternate sides belong to two parametric families of straight lines which are pencils of lines with centers at two points (possibly the point at infinity). The most well-known example of isothetic polygons are rectilinear polygons, and the former term is commonly used as a synonym for the latter one. The term is produced from Greek roots: "iso-" for "equal, same, similar" and "thetos" (position, placement), i,e., the term is supposed to mean "polygon with similarly placed sides". The term was suggested during the early years of the computational geometry. Much emphasis

neg: Fundamental polygon two paired sides can have a common vertex in the final pairing relation, which satisfies similar properties to the original relation. This polygon can in turn be successively modified by reassembling the polygon after cutting it by a diagonal piecewise geodesic segment in its interior. The final polygon has 4"g" equivalent vertices, with sides that are piecewise geodesic. The sides are labelled by the group elements which give the Möbius transformation to the paired side. In order the labelling is so that Γ is generated by the "a" and "b" subject to the single relation Using the theory of intersection

neg: Regular polygon Regular polygon In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. "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

neg: Corresponding sides and corresponding angles Corresponding sides and corresponding angles In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has sequential sides "a", "b", "c", "d", and "e" and the other has sequential sides "v", "w", "x", "y", and "z", and if "b" and "w" are corresponding sides, then side "a" (adjacent to "b") must correspond to either "v" or "x" (both

neg: Equiangular polygon by unit squares, and an equiangular hexagon with integer side lengths may be tiled by unit equilateral triangles. Some but not all equilateral dodecagons may be tiled by a combination of unit squares and equilateral triangles; the rest may be tiled by these two shapes together with rhombi with 30 and 150 degree angles. A cyclic polygon is equiangular if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal and sides 2, 4, ... are equal). Thus if "n" is odd, a cyclic polygon is equiangular if and only if it is

neg: Simple polygon problem definition. Simple polygon In geometry a simple polygon is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pair-wise to form a closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition then being understood to define a polygon in general. The definition given above ensures the following properties: Two edges meeting at a corner are usually required to form an angle that is not straight (180°); otherwise, the collinear line segments will be considered parts of a single side. Mathematicians typically

neg: Regular polygon polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon. 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. The symmetry group of an "n"-sided regular polygon is dihedral group "D"

neg: Regular polygon all "n"-gons with a given perimeter, the one with the largest area is regular. 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, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon. This led to the question being posed: is it possible to construct "all" regular "n"-gons with compass and straightedge? If not, which "n"-gons are constructible and which are not? Carl

neg: Perimeter same length (for example, a rhombus is a 4-sided equilateral polygon). To calculate the perimeter of an equilateral polygon, one must multiply the common length of the sides by the number of sides. A regular polygon may be characterized by the number of its sides and by its circumradius, that is to say, the constant distance between its centre and each of its vertices. The length of its sides can be calculated using trigonometry. If is a regular polygon's radius and is the number of its sides, then its perimeter is A splitter of a triangle is a cevian (a

neg: Polygon partition is considered to be a trapezoid with two horizontal sides one of which is degenerate. For a hole-free polygon with formula_1 sides, a smallest such partition can be found in time formula_5. If the number of trapezoids need not be minimal a trapezoidation can be found in time formula_6, as a by-product of a polygon triangulation algorithm. If the polygon does contain holes, the problem is NP-complete, but a 3-approximation can be found in time formula_7. A "quadrilateralization" or a "quadrangulation" is a partition into quadrilaterals. A recurring characteristic of quadrangulation problems is whether they Steiner point are allowed, i.e.,

neg: Tangential polygon existence criterion above there is only one tangential polygon. But if "n" is even there are an infinitude of them. For example, in the quadrilateral case where all sides are equal we can have a rhombus with any value of the acute angles, and all rhombi are tangential to an incircle. If the "n" sides of a tangential polygon are "a", ..., "a", the inradius (radius of the incircle) is where "K" is the area of the polygon and "s" is the semiperimeter. (Since all triangles are tangential, this formula applies to all triangles.) While all triangles are tangential to

neg: Chiliagon of a chiliagon, he "does not imagine the thousand sides or see them as if they were present" before him – as he does when one imagines a triangle, for example. The imagination constructs a "confused representation," which is no different from that which it constructs of a myriagon (a polygon with ten thousand sides). However, he does clearly understand what a chiliagon is, just as he understands what a triangle is, and he is able to distinguish it from a myriagon. Therefore, the intellect is not dependent on imagination, Descartes claims, as it is able to entertain clear and

neg: Matrix chain multiplication triangulation of a regular polygon. The polygon is oriented such that there is a horizontal bottom side, called the base, which represents the final result. The other "n" sides of the polygon, in the clockwise direction, represent the matrices. The vertices on each end of a side are the dimensions of the matrix represented by that side. With "n" matrices in the multiplication chain there are "n"−1 binary operations and "C" ways of placing parenthesizes, where "C" is the ("n"−1)-th Catalan number. The algorithm exploits that there are also "C" possible triangulations of a polygon with "n"+1 sides. This image

neg: Dual polygon is not a particularly fruitful avenue, as combinatorially, there is a single family of polygons (given by number of sides); geometric duality of polygons is more varied, as are combinatorial dual polyhedra. Dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals. In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter

neg: Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of "n" dimensions is a skew polygon in which every ("n" – 1) consecutive sides (but no "n") belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side (but no three) belongs to one of the faces. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to

neg: Fundamental polygon convex fundamental polygons for compact Riemann surfaces "H"/Γ can be constructed by adapting the method Dirichlet from the Euclidean space to hyperbolic space. Then following Nevanlinna and Jost, the fundamental domain can be modified in steps to yield a non-convex polygon with vertices lying in a single orbit of Γ and piecewise geodesic sides. The pairing relation on the sides is also modified in each of these steps. Each step involves cutting the polygon by a diagonal geodesic segment in the interior of the polygon and reassembling the polygon using one of the Möbius transformations involved in the pairing. No

neg: Schwarz triangle function with all its angles less than is convex, i.e. the straight line joining interior points of the polygon also lies in the interior of the polygon. Since the Poincaré-Klein map preserves the property that angles are less than , a hyperbolic polygon with interior angles less than is carried onto a Euclidean polygon with the same property; the Euclidean polygon is therefore convex and hence, since hyperbolic geodesics are carried onto straight lines, so is the hyperbolic polygon. By a continuity argument, geodesics between points on the sides also lie in the closure of the polygon. A similar convexity result

neg: Aristotle's wheel paradox on the polygon becomes very large, and finds that the little space that is "jumped" by the inner polygon becomes smaller and smaller, writing:Therefore a larger polygon having a thousand sides passes over and measures a straight line equal to its perimeter, while at the same time the smaller one passes an approximately equal line, but one interruptedly composed of a thousand little particles equal to its thousand sides with a thousand little void spaces interposed — for we may call these "void" in relation to the thousand linelets touched by the sides of the polygon.Since the circle is just

neg: Polygon primarily classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Euclidean geometry is assumed throughout. Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: For a non-self-intersecting (simple) polygon with vertices formula_6 the signed area and the Cartesian coordinates of the centroid are given by: where formula_9 is the squared distance between formula_10 and formula_11 and To close the polygon, the first and last vertices are the same, i.e., "x", "y" = "x", "y". The vertices

neg: Tangential polygon some circle, a triangle is called the tangential triangle of a reference triangle if the tangencies of the tangential triangle with the circle are also the vertices of the reference triangle. Tangential polygon In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an "incircle"). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices. All triangles are tangential, as are all

neg: Surface (topology) as a special kind of parametric surface. If "f" is a smooth function from R to R whose gradient is nowhere zero, then the locus of zeros of "f" does define a surface, known as an "implicit surface". If the condition of non-vanishing gradient is dropped, then the zero locus may develop singularities. Each closed surface can be constructed from an oriented polygon with an even number of sides, called a fundamental polygon of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels ("A" with "A", "B" with "B"),

neg: Midpoint at its midpoint. In a regular polygon with an even number of sides, the midpoint of a diagonal between opposite vertices is the polygon's center. The midpoint-stretching polygon of a cyclic polygon (a polygon whose vertices all fall on the same circle) is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the circular arcs between the vertices of . Iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shapes converge to that of a regular polygon. The abovementioned formulas for the midpoint of a segment

neg: Catalan number the triangulation definition of Catalan numbers to establish a relation between "C" and "C". Given a polygon "P" with "n" + 2 sides, first mark one of its sides as the base. If "P" is then triangulated, we can further choose and orient one of its 2"n" + 1 edges. There are (4"n" + 2)"C" such decorated triangulations. Now given a polygon "Q" with "n" + 3 sides, again mark one of its sides as the base. If "Q" is triangulated, we can further mark one of the sides other than the base side. There are ("n" + 2)"C" such

neg: Perimeter triangle problem is the equilateral triangle. In general, the polygon with sides having the largest area and a given perimeter is the regular polygon, which is closer to being a circle than is any irregular polygon with the same number of sides. The word comes from the Greek περίμετρος "perimetros" from περί "peri" "around" and μέτρον "metron" "measure". Perimeter A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path or its length—it can be thought of as the length of the outline of a shape. The perimeter of a circle or

neg: Fundamental polygon the upper half-plane, then one can define the Dirichlet polygon as the set of points Here, "d" is a hyperbolic metric on the upper half-plane. The metric fundamental polygon is more usually called the Dirichlet polygon. In this section, starting from an arbitrary Dirichlet polygon, a description will be given of the method of , elaborated in , for modifying the polygon to a non-convex polygon with 4g equivalent vertices and a canonical pairing on the sides. This treatment is an analytic counterpart of the classical topological classification of orientable 2-dimensional polyhedra presented in . Given a Riemann surface of

neg: Octacontagon be dissected into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular octacontagon", "m"=40, and it can be divided into 780: 20 squares and 19 sets of 40 rhombs. This decomposition is based on a Petrie polygon projection of a 40-cube. An octacontagram is an 80-sided star polygon. There are 15 regular forms given by Schläfli symbols {80/3}, {80/7}, {80/9}, {80/11}, {80/13}, {80/17}, {80/19}, {80/21}, {80/23}, {80/27}, {80/29}, {80/31}, {80/33}, {80/37}, and {80/39}, as well as 24 regular star figures with the same vertex

neg: Hexacontatetragon into "m"("m"-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the "regular hexacontatetragon", "m"=32, and it can be divided into 496: 16 squares and 15 sets of 32 rhombs. This decomposition is based on a Petrie polygon projection of a 32-cube. A hexacontatetragram is a 64-sided star polygon. There are 15 regular forms given by Schläfli symbols {64/3}, {64/5}, {64/7}, {64/9}, {64/11}, {64/13}, {64/15}, {64/17}, {64/19}, {64/21}, {64/23}, {64/25}, {64/27}, {64/29}, {64/31}, as well as 16 compound star figures with the same vertex configuration. Hexacontatetragon In

neg: Apothem because formula_5 Apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent. For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the

neg: Apothem Apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent. For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the slant height

neg: Perimeter to determining perimeters, not only because they are the simplest shapes but also because the perimeters of many shapes are calculated by approximating them with sequences of polygons tending to these shapes. The first mathematician known to have used this kind of reasoning is Archimedes, who approximated the perimeter of a circle by surrounding it with regular polygons. The perimeter of a polygon equals the sum of the lengths of its sides (edges). In particular, the perimeter of a rectangle of width formula_9 and length formula_10 equals formula_11 An equilateral polygon is a polygon which has all sides of the

neg: Equidistant point. This result can be generalised to cyclic polygons: the circumcentre is equidistant from each of the vertices. Likewise, the incentre of a triangle or any other tangential polygon is equidistant from the points of tangency of the polygon's sides with the circle. Every point on a perpendicular bisector of the side of a triangle or other polygon is equidistant from the two vertices at the ends of that side. Every point on the bisector of an angle of any polygon is equidistant from the two sides that emanate from that angle. The center of a rectangle is equidistant from

neg: Equilateral polygon if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides "n" is odd, a tangential polygon is equilateral if and only if it is regular. Viviani's theorem generalizes to equilateral polygons: The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point. The "principal diagonals" of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side "a", there

neg: Tangential polygon only if all of its internal angle bisectors are concurrent. This common point is the "incenter" (the center of the incircle). There exists a tangential polygon of "n" sequential sides "a", ..., "a" if and only if the system of equations has a solution ("x", ..., "x") in positive reals. If such a solution exists, then "x", ..., "x" are the "tangent lengths" of the polygon (the lengths from the vertices to the points where the incircle is tangent to the sides). If the number "n" of sides is odd, then for any given set of sidelengths formula_2 satisfying the

neg: Petrie polygon of Petrie polygons was later extended to semiregular polytopes. The Petrie polygon of the regular polyhedron {"p", "q"} has "h" sides, where The regular duals, {"p","q"} and {"q","p"}, are contained within the same projected Petrie polygon. The regular Kepler–Poinsot polyhedra have hexagonal, {6}, and decagrammic, {10/3}, Petrie polygons. Infinite regular skew polygons (apeirogon) can also be defined as being the Petrie polygons of the regular tilings, having angles of 90, 120, and 60 degrees of their square, hexagon and triangular faces respectively. Infinite regular skew polygons also exist as Petrie polygons of the regular hyperbolic tilings, like the order-7 triangular

neg: Regular polygon (of order 2"n"): "D", "D", "D", ... It consists of the rotations in "C", 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. All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar. An "n"-sided convex regular polygon

neg: Polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red. Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a perfectly regular lattice like above. If is the number of sides in a polygon, the formula for the th -gonal number is or The th -gonal

neg: Equiangular polygon Equiangular polygon In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. The only equiangular triangle is the equilateral triangle. Rectangles, including the square, are the only equiangular quadrilaterals (four-sided figures). For a convex equiangular "n"-gon each internal angle is 180(1-2/n)°; this is the "equiangular polygon theorem". Viviani's theorem holds for equiangular polygons: A rectangle (equiangular quadrilateral) with integer side lengths may be tiled

neg: Hyperbolic geometry or hypercycle, then the triangle has no circumscribed circle. As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent. A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides. In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line. However, in hyperbolic geometry, a regular apeirogon has sides of

neg: Point in polygon the vertex in both cases. If the polygon is specified by its vertices, then this problem is eliminated by checking the y-coordinates of the ray and the ends of the tested polygon side before actual computation of the intersection. In other cases, when polygon sides are computed from other types of data, other tricks must be applied for the numerical robustness of the algorithm. Another algorithm is to compute the given point's winding number with respect to the polygon. If the winding number is non-zero, the point lies inside the polygon. This algorithm is sometimes also known as the nonzero-rule

neg: Golygon vertex and having consecutive sides of length 1, 2, ..., n units. The polygon may be self-crossing. Golygons are a special case of serial-sided isogons. The three-dimensional generalization of a golygon is called a golyhedron–a closed simply-connected solid figure confined to the faces of a cubical lattice and having face areas in the sequence 1, 2, ..., n, for some integer n. Golyhedrons have been found with values of n equal to 32, 15, 12, and 11 (the minimum possible). Golygon A golygon is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons

neg: Elementary mathematics loop to form a closed chain or "circuit". These segments are called its "edges" or "sides", and the points where two edges meet are the polygon's "vertices" (singular: vertex) or "corners". The interior of the polygon is sometimes called its "body". An "n"-gon is a polygon with "n" sides. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. A circle is a simple shape of two-dimensional geometry that is the set of all points in a plane that are at a given distance from a given point, the center.The distance between any of

neg: Two ears theorem has at least two ears. An ear and its two neighbors form a triangle within the polygon that is not crossed by any other part of the polygon. Removing a triangle of this type produces a polygon with fewer sides, and repeatedly removing ears allows any simple polygon to be triangulated. Conversely, if a polygon is triangulated, the weak dual of the triangulation (a graph with one vertex per triangle and one edge per pair of adjacent triangles) will be a tree and each leaf of the tree will form an ear. Since every tree with more than one vertex

neg: Circle is tangent to each of the three sides of the triangle. About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices. A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon. Every regular polygon and every triangle is a tangential polygon. A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon

neg: Straightedge and compass construction mathematicians first attempted straightedge and compass constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and a regular polygon with 3, 4, or 5 sides (or one with twice the number of sides of a given polygon). But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle,

neg: Zonogon a zonogon. Zonogon In geometry, a zonogon is a centrally symmetric convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations. A regular polygon is a zonogon if and only if it has an even number of sides. Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the rectangles, the rhombi, and the parallelograms. The four-sided and six-sided zonogons are parallelogons, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form. Every formula_1-sided

neg: Zonogon Zonogon In geometry, a zonogon is a centrally symmetric convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations. A regular polygon is a zonogon if and only if it has an even number of sides. Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the rectangles, the rhombi, and the parallelograms. The four-sided and six-sided zonogons are parallelogons, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form. Every formula_1-sided zonogon can

neg: Space partitioning least "n" nanometers from any other polygon. This is converted into a geometry query by enlarging a polygon by "n/2" at all sides and query to find all intersecting polygons. The number of components in a space partition plays a central role in some results in probability theory. See Growth function for more details. Common space-partitioning systems include: Suppose the n-dimensional Euclidean space is partitioned by formula_1 hyperplanes that are formula_2-dimensional. What is the number of components in the partition? The largest number of components is attained when the hyperplanes are in general position, i.e, no two are parallel and

neg: Regular polygon (just as there is for a regular polygon). A quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex. A regular polyhedron is a uniform polyhedron which has just one kind of face. The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids. A polyhedron having regular triangles as faces is called a deltahedron. Regular polygon In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or

neg: Concyclic points the sides of a triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, in what is called the Lemoine circle. The van Lamoen circle associated with any given triangle formula_6 contains the circumcenters of the six triangles that are defined inside formula_6 by its three medians. A triangle's circumcenter, its Lemoine point, and its first two Brocard points are concyclic, with the segment from the circumcenter to the Lemoine point being a diameter. A polygon is defined to be cyclic if its vertices are all concyclic. For example, all the vertices

neg: Sutherland–Hodgman algorithm are added to the output list where the subject polygon path crosses the extended clip polygon line. This process is repeated iteratively for each clip polygon side, using the output list from one stage as the input list for the next. Once all sides of the clip polygon have been processed, the final generated list of vertices defines a new single polygon that is entirely visible. Note that if the subject polygon was concave at vertices outside the clipping polygon, the new polygon may have coincident (i.e., overlapping) edges – this is acceptable for rendering, but not for other applications

neg: Polygon triangulation number of ways to triangulate a convex "n"-gon by non-intersecting diagonals is the ("n" − 2)-th Catalan number, which equals formula_1, a solution found by Leonhard Euler. A monotone polygon can be triangulated in linear time with either the algorithm of A. Fournier and D.Y. Montuno, or the algorithm of Godfried Toussaint. One way to triangulate a simple polygon is based on the two ears theorem, as the fact that any simple polygon with at least 4 vertices without holes has at least two 'ears', which are triangles with two sides being the edges of the polygon and the third

neg: Constructible polygon to compass and straightedge construction. More constructions become possible if other tools are allowed. The so-called neusis constructions, for example, make use of a "marked" ruler. The constructions are a mathematical idealization and are assumed to be done exactly. A regular polygon with "n" sides can be constructed with ruler, compass, and angle trisector if and only if formula_3 where "r, s, k" ≥ 0 and where the "p" are distinct Pierpont primes greater than 3 (primes of the form formula_4 Constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge.

neg: Fundamental polygon As such, each side can thus an element of the fundamental group formula_17. In particular, the fundamental group formula_18 has 2"g" generators formula_19, with exactly one defining constraint, The genus of the Riemann surface H/Γ is "g". The area of the standard fundamental polygon is formula_21 where "g" is the genus of the Riemann surface (equivalently, where 4"g" is the number of the sides of the polygon). Since the standard polygon is a representative of H/Γ, the total area of the Riemann surface is equal to the area of the standard polygon. The area formula follows from the Gauss–Bonnet theorem

neg: Internal and external angles Internal and external angles In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than 180°, the polygon is called convex. In contrast, an exterior angle (or ) is an angle formed by one side of a

neg: Skew polygon even-sided n-gonal prisms, alternatingly following an edge of one side polygon, and moving between polygons. For example, on the vertices of a cube. Vertices alternate between top and bottom squares with red edges between sides, and blue edges along each side. In 4 dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike zig-zag skew polygons, skew polygons on double rotations can include an odd-number of sides. The petrie polygons of the regular 4-polytope define regular skew polygons. The Coxeter number for each coxeter group symmetry expresses how many sides a

neg: Incircle and excircles of a triangle polygon with any number of sides that has an inscribed circle—one that is tangent to each side—is called a tangential polygon. Incircle and excircles of a triangle In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three

neg: Petrie polygon tiling, {3,7}: The Petrie polygon for the regular polychora {"p", "q" ,"r"} can also be determined. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and higher. This table represents Petrie polygon projections of 3 regular families (simplex, hypercube, orthoplex), and the exceptional Lie group E which generate semiregular and uniform polytopes for dimensions 4 to 8. Petrie polygon In geometry, a Petrie polygon for a regular polytope of "n" dimensions is a skew polygon in which every ("n" – 1) consecutive sides (but no "n") belongs to one of the facets. The Petrie polygon

neg: Polygram (geometry) a compound polygon. These figures are called regular compound polygons. Polygram (geometry) In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides, so a regular pentagram, {5/2}, has 5 sides, and the regular hexagram, {6/2} or 2{3} has 6 sides divided into two triangles. A regular polygram {p/q} can either be in a set of regular polygons (for gcd(p,q)=1, q>1) or in a set of regular polygon compounds (if gcd(p,q)>1). The polygram names combine a numeral prefix, such as "penta-", with the Greek suffix "-gram" (in this case generating the word "pentagram").

neg: Bicentric polygon line at the points where it crosses the outer circle, continuing from each vertex along another tangent line, and continuing in the same way until the resulting polygonal chain closes up to an "n"-gon. The fact that it will always do so is implied by Poncelet's closure theorem, which more generally applies for inscribed and circumscribed conics. Moreover, given a circumcircle and incircle, each diagonal of the variable polygon is tangent to a fixed circle. Bicentric polygon In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is

neg: Triangle inequality length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths. Consider a quadrilateral whose sides are in a geometric progression and let the sides be , , , . Then the generalized polygon inequality requires that The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, is limited to the range where is the tribonacci constant. This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is

neg: Viviani's theorem the distance from the center to a side. However, the converse does not hold; the non-square parallelogram is a counterexample. The sum of the distances from an interior point to the sides of an equiangular polygon does not depend on the location of the point. A necessary and sufficient condition for a convex polygon to have a constant sum of distances from any interior point to the sides is that there exist three non-collinear interior points with equal sums of distances. The sum of the distances from any point in the interior of a regular polyhedron to the sides is