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This essential trains for: SAT-I, GMAT, Math Kangaroo 7-8, Math Kangaroo 9-10.
Polygons are plane (2D) figures that are formed by placing segments end-to-end.
Polygons can be self-intersecting (a mess!):
In a convex polygon, any two randomly chosen interior points can be connected by a segment that is entirely contained inside the polygon.
In a concave polygon, there is at least a pair of interior points that cannot be connected by a segment that is entirely contained inside the polygon.
A polygon is regular if it all its sides (edges) are congruent and all its interior angles are congruent.
The sum of all the interior angles of a convex polygon is:
Assume the polygon is convex, irregular and has N sides. Simply choose a random point P inside the polygon. Connect this point to all the vertices of the polygon forming N triangles:
We obtain N triangles. The sum of the interior angles of each triangle is 180°. This sum is equal to the sum of the interior angles of the polygon minus the central angle around P of 360° that was added because of the construction:
which simplifies to give the known formula.
This formula does not work for concave or self-intersecting polygons.
The sum of all the exterior angles of a polygon is 360°.
This is because, obviously, the exterior angles must add up to a complete rotation:
An axis of symmetry for a polygon is a line that divides the figure in two figures that are the mirror image of one another.
The perimeter and the area of complicated polygons can be computed by splitting them into simpler figures such as triangles and/or rectangles.