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In geometry, a pentagon (from Greek πέντε (pente) 'five' and γωνία (gonia) 'angle' [1]) 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.
The density of a polygon can also be called its turning number: the sum of the turn angles of all the vertices, divided by 360°. The symmetry group of { p / q } is the dihedral group D p , of order 2 p , independent of q .
The five points of a regular pentagram are golden triangles, [47] as are the ten triangles formed by connecting the vertices of a regular decagon to its center point. [48] Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon.
An n-pointed magic star is a star polygon with Schläfli symbol {n/2} [1] in which numbers are placed at each of the n vertices and n intersections, such that the four numbers on each line sum to the same magic constant. [2] A normal magic star contains the integers from 1 to 2n with no numbers repeated. [3]
The angles of proper spherical triangles are (by convention) less than π, so that < + + < (Todhunter, [1] Art.22,32). In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly π radians.
The pentagram contains ten points (the five points of the star, and the five vertices of the inner pentagon) and fifteen line segments. It is represented by the Schläfli symbol {5/2}. Like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of ...
The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices. Proof: Follows immediately from Ptolemy's theorem: q s = p s + r s ⇒ q = p + r . {\displaystyle qs=ps+rs\Rightarrow q=p+r.}
Convex equilateral pentagon dissected into 3 triangles, which helps to calculate the value of angle δ as a function of α and β. When a convex equilateral pentagon is dissected into triangles, two of them appear as isosceles (triangles in orange and blue) while the other one is more general (triangle in green).