"Octadecagon An octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. A ""regular octadecagon"" has a Schläfli symbol {18} and can be constructed as a quasiregular truncated enneagon, t{9}, which alternates two types of edges. As 18 = 2 × 3, a regular octadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. The ""regular"
"Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ""oktágōnon"", ""eight angles"") is an eight-sided polygon or 8-gon. A ""regular octagon"" has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, t{16}. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a"
"Octacontagon In geometry, an octacontagon (or ogdoëcontagon or 80-gon from Ancient Greek ὁγδοήκοντα, eighty) is an eighty-sided polygon. The sum of any octacontagon's interior angles is 14040 degrees. A ""regular octacontagon"" is represented by Schläfli symbol {80} and can also be constructed as a truncated tetracontagon, t{40}, or a twice-truncated icosagon, tt{20}, or a thrice-truncated decagon, ttt{10}, or a four-fold-truncated pentagon, tttt{5}. One interior angle in a regular octacontagon is 175°, meaning that one exterior angle would be 4°. The area of a regular octacontagon is (with ) and its inradius is The circumradius of a regular octacontagon is Since"
"configuration. Octacontagon In geometry, an octacontagon (or ogdoëcontagon or 80-gon from Ancient Greek ὁγδοήκοντα, eighty) is an eighty-sided polygon. The sum of any octacontagon's interior angles is 14040 degrees. A ""regular octacontagon"" is represented by Schläfli symbol {80} and can also be constructed as a truncated tetracontagon, t{40}, or a twice-truncated icosagon, tt{20}, or a thrice-truncated decagon, ttt{10}, or a four-fold-truncated pentagon, tttt{5}. One interior angle in a regular octacontagon is 175°, meaning that one exterior angle would be 4°. The area of a regular octacontagon is (with ) and its inradius is The circumradius of a regular octacontagon is"
"a sequence of expanded hypercubes: Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ""oktágōnon"", ""eight angles"") is an eight-sided polygon or 8-gon. A ""regular octagon"" has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, t{16}. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers"
"Hexadecagon In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. A ""regular hexadecagon"" is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}. As 16 = 2 (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any"
"Hendecagon In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name ""hendecagon"", from Greek ""hendeka"" ""eleven"" and ""gon–"" ""corner"", is often preferred to the hybrid ""undecagon"", whose first part is formed from Latin ""undecim"" ""eleven"".) A ""regular hendecagon"" is represented by Schläfli symbol {11}. A regular hendecagon has internal angles of 147.27 degrees (=147 formula_1 degrees). The area of a regular hendecagon with side length ""a"" is given by As 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a Pierpont prime, construction"
"of the Virgin"" has 16 sides, elaborating on an eight-sided tower in a previous painting by Pietro Perugino. Hexadecagrams (16-sided star polygons) are included in the Girih patterns in the Alhambra. In the Philippines, in local carnivals (peryahan), Ferris Wheels with maximum of 16 seats or gondolas An octagonal star can be seen as a concave hexadecagon: Hexadecagon In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. A ""regular hexadecagon"" is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a"
"Octadecahedron In geometry, an octadecahedron (or octakaidecahedron) is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron. In chemistry, """"the"" octadecahedron"" commonly refers to a specific structure with C symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [BH]. There are 107,854,282,197,058 topologically distinct ""convex"" octadecahedra, excluding mirror images, having at least 11 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one"
"Decagon In geometry, a decagon is a ten-sided polygon or 10-gon. A ""regular decagon"" has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is {10} and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges. The area of a regular decagon of side length ""a"" is given by: In terms of the apothem ""r"" (see also inscribed figure), the area is: In terms of the circumradius ""R"", the area is: An alternative formula is formula_4 where ""d"" is the distance between parallel"
"Elongated octahedron In geometry, an elongated octahedron is a polyhedron with 8 faces (4 triangular, 4 isosceles trapezoidal), 14 edges, and 8 vertices. A relatd construction is a hexadecahedron, 16 triangular faces, 24 edges, and 10 vertices. Starting with the regular octahedron, it is elongated along one axes, adding 8 new triangles. It has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid. If the sets of coplanar triangles are considered a single isosceles trapezoidal face (a triamond), it has 8 vertices, 14 edges, and 8 faces - 4 triangles and"
"or 255-gon and any regular ""n""-gon with ""2"" times as many sides. Another construction of the regular heptadecagon using straightedge and compass is the following: T. P. Stowell of Rochester, N. Y., responded to Query, by W.E. Heal, Wheeling, Indiana in ""The Analyst"" in the year 1874: """"To construct a regular polygon of seventeen sides in a circle."" ""Draw the radius CO at right-angles to the diameter AB: On OC and OB, take OQ equal to the half, and OD equal to the eighth part of the radius: Make DE and DF each equal to DQ and EG and FH"
"Hexagon In geometry, a hexagon (from Greek ἕξ ""hex"", ""six"" and γωνία, ""gonía"", ""corner, angle"") is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. A ""regular hexagon"" has Schläfli symbol {1} and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius"
"side edges of a octagonal antiprism with the same D, [2,16] symmetry, order 32. The octagrammic antiprism, s{2,16/3} and octagrammic crossed-antiprism, s{2,16/5} also have regular skew octagons. The regular hexadecagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including: A hexadecagram is a 16-sided star polygon, represented by symbol {16/n}. There are three regular star polygons, {16/3}, {16/5}, {16/7}, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds: {16/2} is reduced to 2{8} as two octagons, {16/4} is reduced to 4{4} as four squares and {16/6}"
"Dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol {12} and can be constructed as a truncated hexagon, t{6}, or a twice-truncated triangle, tt{3}. The internal angle at each vertex of a regular dodecagon is 150°. The area of a regular dodecagon of side length ""a"" is given by: And in terms of the apothem ""r"" (see"
"two kinds of polygon. The regular octadecagon can tessellate the plane with concave hexagonal gaps. And another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a truncated hexagonal tiling, and the second the truncated trihexagonal tiling. An octadecagram is an 18-sided star polygon, represented by symbol {18/n}. There are two regular star polygons: {18/5} and {18/7}, using the same points, but connecting every fifth or seventh points. There are also five compounds: {18/2} is reduced to 2{9} or two enneagons, {18/3} is reduced to 3{6} or three hexagons, {18/4} and {18/8} are reduced to 2{9/2}"
"Triacontadigon In geometry, a triacontadigon (or triacontakaidigon) or 32-gon is a thirty-two-sided polygon. In Greek, the prefix triaconta- means 30 and di- means 2. The sum of any triacontadigon's interior angles is 5400 degrees. An older name is tricontadoagon. Another name is icosidodecagon, suggesting a (20 and 12)-gon, in parallel to the 32-faced icosidodecahedron, which has 20 triangles and 12 pentagons. The ""regular triacontadigon"" can be constructed as a truncated hexadecagon, t{16}, a twice-truncated octagon, tt{8}, and a thrice-truncated square. A truncated triacontadigon, t{32}, is a hexacontatetragon, {64}. One interior angle in a regular triacontadigon is 168°, meaning that one"
"Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices. Since it has eight faces, it is an octahedron. However, the term ""octahedron"" is primarily used to refer to the ""regular octahedron"", which has eight triangular faces. Because of the ambiguity of the term ""octahedron"" and the dissimilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long hexagonal prism. If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a"
"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"
"Tetracontaoctagon In geometry, a tetracontaoctagon (or tetracontakaioctagon) or 48-gon is a forty-eight sided polygon. The sum of any tetracontaoctagon's interior angles is 8280 degrees. The ""regular tetracontaoctagon"" is represented by Schläfli symbol {48} and can also be constructed as a truncated icositetragon, t{24}, or a twice-truncated dodecagon, tt{12}, or a thrice-truncated hexagon, ttt{6}, or a fourfold-truncated triangle, tttt{3}. One interior angle in a regular tetracontaoctagon is 172°, meaning that one exterior angle would be 7°. The area of a regular tetracontaoctagon is: (with ""t"" = edge length) The tetracontaoctagon appeared in Archimedes' polygon approximation of pi, along with the hexagon"
"configuration. Tetracontaoctagon In geometry, a tetracontaoctagon (or tetracontakaioctagon) or 48-gon is a forty-eight sided polygon. The sum of any tetracontaoctagon's interior angles is 8280 degrees. The ""regular tetracontaoctagon"" is represented by Schläfli symbol {48} and can also be constructed as a truncated icositetragon, t{24}, or a twice-truncated dodecagon, tt{12}, or a thrice-truncated hexagon, ttt{6}, or a fourfold-truncated triangle, tttt{3}. One interior angle in a regular tetracontaoctagon is 172°, meaning that one exterior angle would be 7°. The area of a regular tetracontaoctagon is: (with ""t"" = edge length) The tetracontaoctagon appeared in Archimedes' polygon approximation of pi, along with the"
"hebesphenorotunda. In addition, some uniform star polyhedra are also octadecahedra: Octadecahedron In geometry, an octadecahedron (or octakaidecahedron) is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron. In chemistry, """"the"" octadecahedron"" commonly refers to a specific structure with C symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [BH]. There are 107,854,282,197,058 topologically distinct ""convex"" octadecahedra, excluding mirror images, having at least 11 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces"
"8-8 duoprism In geometry of 4 dimensions, a 8-8 duoprism or octagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons. It has 64 vertices, 128 edges, 80 faces (64 squares, and 16 octagons), in 16 octagonal prism cells. It has Coxeter diagram , and symmetry , order 512. The uniform ""8-8 duoprism"" can be constructed from [8]×[8] or [4]×[4] symmetry, order 256 or 64, with extended symmetry doubling these with a 2-fold rotation that maps the two orientations of prisms together. These can be expressed by 4 permutations of uniform coloring of the"
"Enneadecahedron A enneadecahedron (or enneakaidecahedron) is a polyhedron with 19 faces. No enneadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of an enneadecahedron, for example the octadecagonal (18-sided) pyramid, or the heptadecagonal (17-sided) prism; the latter is the only convex enneadecahedron with all regular polygonal faces. There are many topologically distinct ""convex"" enneadecahedra, excluding mirror images, having at least 12 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges"
"sides, or the height when the decagon stands on one side as base, or the diameter of the decagon's inscribed circle. By simple trigonometry, and it can be written algebraically as The side of a regular decagon inscribed in a unit circle is formula_7, where ""ϕ"" is the golden ratio, formula_8. As 10 = 2 × 5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon. An alternative (but similar) method is as follows: The length ratio of two inequal edges"
"gaps. This represents a section of a tetrahedral-octahedral honeycomb. The ""elongated octahedron"" can thus be used with the tetrahedron as a space-filling honeycomb. Elongated octahedron In geometry, an elongated octahedron is a polyhedron with 8 faces (4 triangular, 4 isosceles trapezoidal), 14 edges, and 8 vertices. A relatd construction is a hexadecahedron, 16 triangular faces, 24 edges, and 10 vertices. Starting with the regular octahedron, it is elongated along one axes, adding 8 new triangles. It has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid. If the sets of coplanar"
"Enneacontahexagon In geometry, an enneacontahexagon (or enneacontakaihexagon) or 96-gon is a ninety-six-sided polygon. The sum of any enneacontahexagons's interior angles is 16920 degrees. The ""regular enneacontahexagon"" is represented by Schläfli symbol {96} and can also be constructed as a truncated tetracontaoctagon, t{48}, or a twice-truncated icositetragon, tt{24}, or a thrice-truncated dodecagon, ttt{12}, or a fourfold-truncated hexagon, tttt{6}, or a fivefold-truncated triangle, }. One interior angle in a regular enneacontahexagon is 176°, meaning that one exterior angle would be 3°. The area of a regular enneacontahexagon is: (with ""t"" = edge length) The enneacontahexagon appeared in Archimedes' polygon approximation of pi,"
"4 triamonds . This construction has been called a triamond stretched octahedron. Another interpretation can represent this solid as a hexahedron, by considering pairs of trapezoids as a folded regular hexagon. It will have 6 faces (4 triangles, and 2 hexagons), 12 edges, and 8 vertices. It could also be seen as a ""folded tetrahedron"" also seeing pairs of end triangles as a folded rhombus. It would have 8 vertices, 10 edges, and 4 faces. The Cartesian coordinates of the 8 vertices of an ""elongated octahedron"", elongated in the x-axis, with edge length 2 are: The 2 extra vertices of"
"Tridecagon In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon. A ""regular tridecagon"" is represented by Schläfli symbol {13}. The measure of each internal angle of a regular tridecagon is approximately 152.308 degrees, and the area with side length ""a"" is given by As 13 is a Pierpont prime but not a Fermat prime, the regular tridecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector. The following is an animation from a ""neusis construction"" of a regular tridecagon with radius of circumcircle formula_2 according to Andrew M."
"of 8 isosceles triangles, leading to the result: for an octagon of side ""a"". The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are: 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 octagon"", ""m""=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6"
"Hexadecahedron A hexadecahedron (or hexakaidecahedron) is a polyhedron with 16 faces. No hexadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a hexadecahedron, for example the pentadecagonal pyramid, tetradecagonal prism and heptagonal antiprism. There are 387,591,510,244 topologically distinct ""convex"" hexadecahedra, excluding mirror images, having at least 10 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) There are 302,404 self-dual hexadecahedron, 1476"
"Tetradecagon In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon. A ""regular tetradecagon"" has Schläfli symbol {14} and can be constructed as a quasiregular truncated heptagon, t{7}, which alternates two types of edges. The area of a regular tetradecagon of side length ""a"" is given by As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis with use of the angle trisector, or with a marked ruler, as it in the following two examples showned. The animation below gives an approximation of about 0.05°"
"Nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ""nonagon"" is a prefix hybrid formation, from Latin (""nonus"", ""ninth"" + ""gonon""), used equivalently, attested already in the 16th century in French ""nonogone"" and in English from the 17th century. The name ""enneagon"" comes from Greek ""enneagonon"" (εννεα, ""nine"" + γωνον (from γωνία = ""corner"")), and is arguably more correct, though less common than ""nonagon"". A ""regular nonagon"" is represented by Schläfli symbol {9} and has internal angles of 140°. The area of a regular nonagon of side length ""a"" is given by"