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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 ...
If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side. [1]: pp. 261–264
For a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is ...
Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple ...
The region inside the polygon (its interior) forms a bounded set [2] topologically equivalent to an open disk by the Jordan–Schönflies theorem, [10] with a finite but nonzero area. [11] The polygon itself is topologically equivalent to a circle, [12] and the region outside (the exterior) is an unbounded connected open set, with infinite area ...
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 ...