In mathematics what is the sum of the internal angles of a pentagon?
"of δ can be seen from the figure: Simplifying, δ is obtained as function of α and β: The remaining angles of the pentagon can be found geometrically: The remaining angles of the orange and blue triangles are readily found by noting that two angles of an isosceles triangle are equal while all three angles sum to 180°. Then formula_7 and the two remaining angles of the green triangle can be found from four equations stating that the sum of the angles of the pentagon is 540°, the sum of the angles of the green triangle is 180°, the angle"
"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"
"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"
"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"
"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"
"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"
"This decomposition is based on a Petrie polygon projection of a 25-cube. A pentacontagram is a 50-sided star polygon. There are 9 regular forms given by Schläfli symbols {50/3}, {50/7}, {50/9}, {50/11}, {50/13}, {50/17}, {50/19}, {50/21}, and {50/23}, as well as 16 compound star figures with the same vertex configuration. 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"
"polygon whose angles are all . To find the number of sides this polygon has, the result is , which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons. There are 15 classes of pentagons that can monohedrally tile the plane. None of the pentagons have any symmetry in general, although some have special cases with mirror symmetry. 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"
"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"
"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"
"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"
"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"
"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"
"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"
"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"
"as well as 15 compound star figures with the same vertex configuration. Tetracontadigon In geometry, a tetracontadigon (or tetracontakaidigon) or 42-gon is a forty-two-sided polygon. (In Greek, the prefix tetraconta- means 40 and di- means 2.) The sum of any tetracontadigon's interior angles is 7200 degrees. The ""regular tetracontadigon"" can be constructed as a truncated icosihenagon, t{21}. One interior angle in a regular tetracontadigon is 171°, meaning that one exterior angle would be 8°. The area of a regular tetracontadigon is (with ) and its inradius is The circumradius of a regular tetracontadigon is Since 42 = 2 × 3"
"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"
"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 ="
"Pentagrammic prism In geometry, the pentagrammic prism is one in an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. This polyhedron is identified with the indexed name U as a uniform polyhedron. It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Note that the pentagram face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered interior or exterior depending on how interior is defined. One definition of interior is the"
"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"
"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"
"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"
"a square to a height that the new triangles of the raised tetrahedra become coplanar to the triangles which do share an edge with a square. The result is the pentagonal icositetrahedron. Denote the tribonacci constant by ""t"", approximately 1.8393. (See snub cube for a geometric explanation of the tribonacci constant.) Then the pentagonal faces have four angles of cos () ≈ 114.8° and one angle of cos (2 − ""t"") ≈ 80.75°. The pentagon has three short edges of unit length each, and two long edges of length ≈ 1.42. The acute angle is between the two long edges."
"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,"
"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"
"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"
"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"
"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."
"geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small. The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy formula_1. The Pythagorean result"
"Triacontatetragon In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon. The sum of any triacontatetragon's interior angles is 5760 degrees. A ""regular triacontatetragon"" is represented by Schläfli symbol {34} and can also be constructed as a truncated 17-gon, t{17}, which alternates two types of edges. One interior angle in a regular triacontatetragon is (2880/17)°, meaning that one exterior angle would be (180/17)°. The area of a regular triacontatetragon is (with ) and its inradius is The factor formula_3 is a root of the equation formula_4. The circumradius of a regular triacontatetragon is As 34 = 2 ×"
"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"
"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"
"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"