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It is always possible to partition a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polygons as possible is described by Chazelle & Dobkin (1985). [5] A triangle can never be concave, but there exist concave polygons with n sides for any n > 3.
Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, [1]: p. xi and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.
The side view is an isosceles trapezoid. In first-angle projection, the front view is pushed back to the rear wall, and the right side view is pushed to the left wall, so the first-angle symbol shows the trapezoid with its shortest side away from the circles.
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.
A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol {2}. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated digon, t{2} is a square, {4}.
Polygons with only one concave vertex can always be fan triangulated, as long as the diagonals are drawn from the concave vertex. It can be known if a polygon can be fan triangulated by solving the Art gallery problem, in order to determine whether there is at least one vertex that is visible from every point in the polygon.
The 42 possible triangulations for a convex heptagon (7-sided convex polygon). This number is given by the 5th Catalan number . It is trivial to triangulate any convex polygon in linear time into a fan triangulation , by adding diagonals from one vertex to all other non-nearest neighbor vertices.
Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least four, an equilateral polygon does not need to be a convex polygon : it could be concave or even self-intersecting .