Polygons

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: