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In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star .
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is (n − 2) × π radians or (n − 2) × 180 degrees. This is because any simple n-gon ( having n sides ) can be considered to be made up of (n − 2) triangles ...
The regular hexagon has D 6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D 6), 2 dihedral: (D 3, D 2), 4 cyclic: (Z 6, Z 3, Z 2, Z 1) and the trivial (e) These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order. [4] r12 is full symmetry, and a1 is no ...
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.
The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. [ 64 ] As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the ...
The regular hexadecagon has Dih 16 symmetry, order 32. There are 4 dihedral subgroups: Dih 8, Dih 4, Dih 2, and Dih 1, and 5 cyclic subgroups: Z 16, Z 8, Z 4, Z 2, and Z 1, the last implying no symmetry. On the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1.
The equidissection problem concerns the subdivision of polygons into triangles that all have equal areas. In this context, the spectrum of a polygon is the set of numbers such that the polygon has an equidissection into equal-area triangles. Because of its symmetry, the spectrum of a kite contains all even integers.