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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.
In several high school treatments of geometry, the term "exterior angle theorem" has been applied to a different result, [1] namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel ...
The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR 2 where R is the circumradius. [4]: p. 73 The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR 2 − 1 / 4 ns 2, where s is the side length and R is the ...
The supplement of an interior angle is called an exterior angle; that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal.
Every finite set of points in the plane that does not lie on a single line can be connected to form the vertices of a simple polygon (allowing 180° angles); for instance, one such polygon is the solution to the traveling salesperson problem. [36] Connecting points to form a polygon in this way is called polygonalization. [37]
One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°.. The area of a regular icositetragon is: (with t = edge length) = = (+ + +).
A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D 3 symmetry. A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle, {3}.
Exterior angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem. One can also consider the sum of all three exterior angles, that equals to 360° [9] in the Euclidean case (as for any convex polygon), is less than 360° in the spherical case, and is greater than 360° in the hyperbolic case.