"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"
"36, 40, 45, 60, 72, 90, 120, 180, 360}. 360-gon In geometry, a 360-gon (triacosihexacontagon or triacosihexecontagon) is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees. A ""regular 360-gon"" is represented by Schläfli symbol {360} and also can be constructed as a truncated 180-gon, t{180}, or a twice-truncated enneacontagon, tt{90}, or a thrice-truncated tetracontapentagon, ttt{45}. One interior angle in a regular 360-gon is 179°, meaning that one exterior angle would be 1°. The area of a regular 360-gon is (with ) and its inradius is The circumradius of a regular 120-gon is Since"
"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"
"Hectogon In geometry, a hectogon or hecatontagon or 100-gon is a hundred-sided polygon. The sum of any hectogon's interior angles is 17640 degrees. A ""regular hectogon"" is represented by Schläfli symbol {100} and can be constructed as a truncated pentacontagon, t{50}, or a twice-truncated icosipentagon, tt{25}. One interior angle in a regular hectogon is 176°, meaning that one exterior angle would be 3°. The area of a regular hectogon is (with ) and its inradius is The circumradius of a regular hectogon is Because 100 = 2 × 5, the number of sides contains a ""repeated"" Fermat prime (the number"
"Icosagon In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees. The ""regular icosagon"" has Schläfli symbol {20}, and can also be constructed as a truncated decagon, t{10}, or a twice-truncated pentagon, tt{5}. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°. The area of a regular icosagon with edge length ""t"" is In terms of the radius ""R"" of its circumcircle, the area is since the area of the circle is formula_3 the regular icosagon fills approximately 98.36% of its circumcircle."
"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"
"360-gon In geometry, a 360-gon (triacosihexacontagon or triacosihexecontagon) is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees. A ""regular 360-gon"" is represented by Schläfli symbol {360} and also can be constructed as a truncated 180-gon, t{180}, or a twice-truncated enneacontagon, tt{90}, or a thrice-truncated tetracontapentagon, ttt{45}. One interior angle in a regular 360-gon is 179°, meaning that one exterior angle would be 1°. The area of a regular 360-gon is (with ) and its inradius is The circumradius of a regular 120-gon is Since 360 = 2 × 3 × 5, a regular"
"Enneacontagon In geometry, an enneacontagon or enenecontagon or 90-gon (from Ancient Greek ἑννενήκοντα, ninety) is a ninety-sided polygon. The sum of any enneacontagon's interior angles is 15840 degrees. A ""regular enneacontagon"" is represented by Schläfli symbol {90} and can be constructed as a truncated tetracontapentagon, t{45}, which alternates two types of edges. One interior angle in a regular enneacontagon is 176°, meaning that one exterior angle would be 4°. The area of a regular enneacontagon is (with ) and its inradius is The circumradius of a regular enneacontagon is Since 90 = 2 × 3 × 5, a regular enneacontagon"
"and {100/49}, as well as 30 regular star figures with the same vertex configuration. Hectogon In geometry, a hectogon or hecatontagon or 100-gon is a hundred-sided polygon. The sum of any hectogon's interior angles is 17640 degrees. A ""regular hectogon"" is represented by Schläfli symbol {100} and can be constructed as a truncated pentacontagon, t{50}, or a twice-truncated icosipentagon, tt{25}. One interior angle in a regular hectogon is 176°, meaning that one exterior angle would be 3°. The area of a regular hectogon is (with ) and its inradius is The circumradius of a regular hectogon is Because 100 ="
"ordinary convex and concave polygons ""k"" = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution walking around the perimeter. 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"
"Triacontagon In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees. The ""regular triacontagon"" is a constructible polygon, by an edge-bisection of a regular pentadecagon, and can also be constructed as a truncated pentadecagon, t{15}. A truncated triacontagon, t{30}, is a hexacontagon, {60}. One interior angle in a regular triacontagon is 168°, meaning that one exterior angle would be 12°. The triacontagon is the largest regular polygon whose interior angle is the sum of the interior angles of smaller polygons: 168° is the sum of the interior angles of the"
"Pentagon In geometry, a pentagon (from the Greek πέντε ""pente"" and γωνία ""gonia"", meaning ""five"" and ""angle"") is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ""regular pentagon"" (or ""star pentagon"") is called a pentagram. A ""regular pentagon"" has Schläfli symbol {5} and interior angles are 108°. A ""regular pentagon"" has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its"
"Tetracontagon In geometry, a tetracontagon or tessaracontagon is a forty-sided polygon or 40-gon. The sum of any tetracontagon's interior angles is 6840 degrees. A ""regular tetracontagon"" is represented by Schläfli symbol {40} and can also be constructed as a truncated icosagon, t{20}, which alternates two types of edges. Furthermore, it can also be constructed as a twice-truncated decagon, tt{10}, or a thrice-truncated pentagon, ttt{5}. One interior angle in a regular tetracontagon is 171°, meaning that one exterior angle would be 9°. The area of a regular tetracontagon is (with ) and its inradius is The factor formula_3 is a root"
"simple polygon and a line extended from an adjacent side. The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(""n""–2""k"")° where ""n"" is the number of vertices and the non-negative number ""k"" is the number of total revolutions of 360° one undergoes walking around the perimeter of the polygon. In other words, 360""k""° represents the sum of all the exterior angles. For example, for"
"Pentacontagon In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon. The sum of any pentacontagon's interior angles is 8640 degrees. A ""regular pentacontagon"" is represented by Schläfli symbol {50} and can be constructed as a quasiregular truncated icosipentagon, t{25}, which alternates two types of edges. One interior angle in a regular pentacontagon is 172°, meaning that one exterior angle would be 7°. The area of a regular pentacontagon is (with ) and its inradius is The circumradius of a regular pentacontagon is Since 50 = 2 × 5, a regular pentacontagon is not constructible using a compass"
"as 44 compound star figures with the same vertex configuration. 120-gon In geometry, a 120-gon is a polygon with 120 sides. The sum of any 120-gon's interior angles is 21240 degrees. Alternative names include dodecacontagon and hecatonicosagon. A ""regular 120-gon"" is represented by Schläfli symbol {120} and also can be constructed as a truncated hexacontagon, t{60}, or a twice-truncated triacontagon, tt{30}, or a thrice-truncated pentadecagon, ttt{15}. One interior angle in a regular 120-gon is 177°, meaning that one exterior angle would be 3°. The area of a regular 120-gon is (with ) and its inradius is The circumradius of a"
"figures, which are also called ""m""-Archimedean tilings. For edge-to-edge Euclidean tilings, the internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular -gon has internal angle formula_1 degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a ""species"" of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one ""types"" of vertex. Only eleven of these can occur in a uniform tiling of regular polygons, given in previous sections. In particular, if three polygons meet at a"
"Heptacontagon In geometry, a heptacontagon (or hebdomecontagon from Ancient Greek ἑβδομήκοντα, seventy) or 70-gon is a seventy-sided polygon. The sum of any heptacontagon's interior angles is 12240 degrees. A ""regular heptacontagon"" is represented by Schläfli symbol {70} and can also be constructed as a truncated triacontapentagon, t{35}, which alternates two types of edges. One interior angle in a regular heptacontagon is 174°, meaning that one exterior angle would be 5°. The area of a regular heptacontagon is (with ) and its inradius is The circumradius of a regular heptacontagon is Since 70 = 2 × 5 × 7, a regular"
"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,"
"Hexacontagon In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon. The sum of any hexacontagon's interior angles is 10440 degrees. A ""regular hexacontagon"" is represented by Schläfli symbol {60} and also can be constructed as a truncated triacontagon, t{30}, or a twice-truncated pentadecagon, tt{15}. A truncated hexacontagon, t{60}, is a 120-gon, {120}. One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°. The area of a regular hexacontagon is (with ) and its inradius is The circumradius of a regular hexacontagon is This means that the trigonometric functions of π/60"
"120-gon In geometry, a 120-gon is a polygon with 120 sides. The sum of any 120-gon's interior angles is 21240 degrees. Alternative names include dodecacontagon and hecatonicosagon. A ""regular 120-gon"" is represented by Schläfli symbol {120} and also can be constructed as a truncated hexacontagon, t{60}, or a twice-truncated triacontagon, tt{30}, or a thrice-truncated pentadecagon, ttt{15}. One interior angle in a regular 120-gon is 177°, meaning that one exterior angle would be 3°. The area of a regular 120-gon is (with ) and its inradius is The circumradius of a regular 120-gon is This means that the trigonometric functions of"
"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"
"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"
"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"
"the sum of the interior angles is π(""n"" − 2) radians or 180(""n"" − 2) degrees. Because more than two lines in a polytope can intersect in the same space, the number of vertex angles from any given vertex is not always one. Because of this, it is sometimes necessary to specify not only the vertex, but also the specific line segments that make up the angle. Vertex angle In geometry, a vertex angle is an angle associated with a vertex of an n-dimensional polytope. Often a vertex angle of a polygon is referred to simply as an angle of"
"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"
"eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° (formula_1 radians). The central angle is 45° (formula_2 radians). The area of a regular octagon of side length ""a"" is given by In terms of the circumradius ""R"", the area is In terms of the apothem ""r"" (see also inscribed figure), the area is These last two coefficients bracket the value of pi, the area of the unit circle. The area can also be expressed as where"
"Icositetragon In geometry, an icositetragon (or icosikaitetragon or tetracosagon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees. The ""regular icositetragon"" is represented by Schläfli symbol {24} and can also be constructed as a truncated dodecagon, t{12}, or a twice-truncated hexagon, tt{6}, or thrice-truncated triangle, ttt{3}. One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°. The area of a regular icositetragon is: (with ""t"" = edge length) The icositetragon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), tetracontaoctagon (48-gon),"
"Hexacontatetragon In geometry, a hexacontatetragon (or hexacontakaitetragon) or 64-gon is a sixty-four-sided polygon. (In Greek, the prefix hexaconta- means 60 and tetra- means 4.) The sum of any hexacontatetragon's interior angles is 11160 degrees. The ""regular hexacontatetragon"" can be constructed as a truncated triacontadigon, t{32}, a twice-truncated hexadecagon, tt{16}, a thrice-truncated octagon, ttt{8}, a fourfold-truncated square, tttt{4}, and a fivefold-truncated digon, }. One interior angle in a regular hexacontatetragon is 174°, meaning that one exterior angle would be 5°. The area of a regular hexacontatetragon is (with ) and its inradius is The circumradius of a regular hexacontatetragon is Since"
"Equilateral pentagon In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees. In contrast, the regular pentagon is unique, because it is equilateral and moreover it is equiangular (its five angles are equal; the measure is 108 degrees). Four intersecting equal circles arranged in a closed chain"