Polygons
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Properties of polygons
A polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or circuit.
The segments of a polygonal circuit are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners.
Polygons are primarily classified by the number of sides. A simple polygon is one which does not intersect itself.
Convex polygon - A polygon is convex if any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints.
Concave polygon - A concave polygon is non-convex. There is at least one interior angle greater than 180°.
Self-intersecting polygon - the boundary of the polygon crosses itself.
Cyclic polygons - polygons with all corners lie on a single circle, called the circumcircle.
Equilateral polygons - polygons with all edges are of the same length. The polygon need not be convex.
Equiangular polygons - polygons with all corner angles equal.
Regular polygon - polygon that is both equilateral and equiangular and equivalently, it is both cyclic and equilateral.
Polygon Names
3 sides - triangle (or trigon)
4 sides - quadrilateral (or tetragon)
5 sides - pentagon
6 sides - hexagon 6
7 sides - heptagon (or septagon)
8 sides - octagon
9 sides - nonagon (or enneagon)
10 sides - decagon
11 sides - hendecagon (or undecagon)
12 sides - dodecagon (or duodecagon)
13 sides - tridecagon (or triskaidecagon)
14 sides - tetradecagon (or tetrakaidecagon)
15 sides - pentadecagon (or pentakaidecagon)
16 sides - hexadecagon (or hexakaidecagon)
17 sides - heptadecagon (or heptakaidecagon)
18 sides - octadecagon (or octakaidecagon)
19 sides - enneadecagon (or enneakaidecagon)
20 sides - icosagon
24 sides - icositetragon (or icosikaitetragon)
30 sides - triacontagon
40 sides - tetracontagon (or tessaracontagon)
50 sides - pentacontagon (or pentecontagon)
60 sides - hexacontagon (or hexecontagon)
70 sides - heptacontagon (or hebdomecontagon)
80 sides - octacontagon (or ogdoëcontagon)
90 sides - enneacontagon (or enenecontagon)
100 sides - hectogon (or hecatontagon)
Angles and diagonals
Each corner of a polygon has an interior and exterior angles. The exterior angle is the supplementary angle to the interior angle.
The sum of the interior angles of a polygon of n sides is:
The sum of the exterior angles for concave polygons is 360°.
The number of diagonals is:
Regular polygons
A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
The distance R from the center of polygon to its vertex is the radius of the circumscribed circle (circumcircle) and the distance r from the center of polygon to the midpoint of the sides, called the apothem, is the radius of the inscribed circle (incircle).
The angle θ subtended by the side of polygon of n sides from its center is:
where n is the number of sides.
The apothem r and circumradius R is:
The area A of regular polygon of n sides is:
The area between the circumcircle and incircle is: