"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"
"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"
"72,72,36 degrees. From trigonometry, we know that the cosine of twice 18 degrees is 1 minus twice the square of the sine of 18 degrees, and this reduces to the desired result with simple quadratic arithmetic. The regular pentagon according to the golden ratio, dividing a line segment by exterior division A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his ""Elements"" circa 300 BC. A direct method using degrees follows: After forming a regular convex pentagon,"
"if one joins the non-adjacent corners (drawing the diagonals of the pentagon), one obtains a pentagram, with a smaller regular pentagon in the center. Or if one extends the sides until the non-adjacent sides meet, one obtains a larger pentagram. The accuracy of this method depends on the accuracy of the protractor used to measure the angles. The ""regular pentagon"" has Dih symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih, and 2 cyclic group symmetries: Z, and Z. These 4 symmetries can be seen in 4 distinct symmetries on the pentagon."
"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"
"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"
"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 ="
"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"
"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"
"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"
"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"
"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"
"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"
"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"
"is denoted by its Schläfli symbol {""n""}. For ""n"" < 3, we have two degenerate cases: In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc. For a regular convex ""n""-gon, each interior angle has a measure of: and each exterior angle (i.e., supplementary to the interior angle) has a measure of formula_5 degrees, with the sum of the exterior angles equal to 360 degrees or"
"are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal hexecontahedron. The faces are irregular pentagons with two long edges and three short edges. The ratio of edge lengths is about 1:1.7498525667362. The faces have four obtuse angles of about 118.13662 degrees each, and one acute angle (between the two long edges) of about 67.45351 degrees. The dihedral angle is about 153.178732558 degrees between all faces. Note that the face"
"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"
"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"
"Pentadecagon In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. A ""regular pentadecagon"" is represented by Schläfli symbol {15}. A regular pentadecagon has interior angles of 156°, and with a side length ""a"", has an area given by <br>A regular triangle, decagon, and pentadecagon can completely fill a plane vertex. As 15 = 3 × 5, a regular pentadecagon is constructible using compass and straightedge: The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's ""Elements"". Comparison the construction according Euclid in this image:"
"Pentadecagon In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. A ""regular pentadecagon"" is represented by Schläfli symbol {15}. A regular pentadecagon has interior angles of 156°, and with a side length ""a"", has an area given by <br>A regular triangle, decagon, and pentadecagon can completely fill a plane vertex. As 15 = 3 × 5, a regular pentadecagon is constructible using compass and straightedge: The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's ""Elements"". Comparison the construction according Euclid in this image:"
"regular pentagon's diagonal is times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, ""Thomson problem""). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/ ≅ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory"
"Pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices. If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a ""truncated pentagonal hosohedron"", represented by Schläfli symbol t{2,5}. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product {5}x{}."
"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."
"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"
"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"
"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"
"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"
"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"
"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"
"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"
"the geometry of pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is , as the four-color illustration shows. The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is . The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons. The golden ratio properties of a regular pentagon"
"Degree (angle) A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians. The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is"
"a simple truncation of a decagram gives t{10/3}={20/3}. The regular icosagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in orthogonal projections in Coxeter planes: It is also the Petrie polygon for the icosahedral 120-cell, small stellated 120-cell, great icosahedral 120-cell, and great grand 120-cell. 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°,"