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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 sum of all the internal angles of a simple polygon is π(n−2) radians or 180(n–2) degrees, where n is the number of sides. The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
Case 2: two sides and an included angle given (SAS). The cosine rule gives a and then we are back to Case 1. Case 3: two sides and an opposite angle given (SSA). The sine rule gives C and then we have Case 7. There are either one or two solutions. Case 4: two angles and an included side given (ASA). The four-part cotangent formulae for sets ...
Then ,, 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 is the sum of its three components, and the angle is the sum of its two components.
For a regular polygon with 10,000 sides (a myriagon) the internal angle is 179.964°. As the number of sides increases, 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 ...
A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side , then also the presentation is succeed by extension one side and it generates a segment, here F E 2 ¯ , {\displaystyle {\overline {FE_{2}}}{\text{,}}} which is divided according to the golden ratio:
The external angle is positive at a convex vertex or negative at a concave vertex. For every simple polygon, the sum of the external angles is (one full turn, 360°). Thus the sum of the internal angles, for a simple polygon with sides is (). [14] A triangulated polygon with 11 vertices: 11 sides and 8 diagonals form 9 triangles. Every simple ...