Introduction to Polygons
As we saw in the Two- and Three-Dimensional Figures section, closed shapes with three or more sides are called Polygons.
More specifically, polygons are coplanar closed shapes that are formed by straight line segments (sides). Some polygons you’ve probably familiar with are rectangles, squares, and triangles. Circles, for example, are not polygons, since they are not formed by straight lines.
A polygon can be convex, where no points of any of the lines are in the interior of the shape, or concave, where some of the lines are in the interior. I like to identify concave polygons by looking for a “cave” or indentation in the shape.
Here is an example of a concave polygon, which has $ 5$ sides (pentagon). See how there’s a “cave” or indentation on the left-hand side? 
. Here is an example of a convex pentagon; notice any line segment between any two points on the figure would be either inside the figure or on the boundary: 
.
Note that this polygon has $ 5$ congruent sides and $ 5$ congruent angles; it is both equilateral, meaning that all sides are the same lengths, and equiangular, meaning that all angles are congruent. Thus, it is a regular polygon.
Here are the names of polygons. Notice also that a larger number of sides is just called a “$ n$n”-gon. For example, a polygon with $ 15$ sides is a $ 15$-gon.
| Number of Sides | Polygon | Number of Sides | Polygon | |
| $ 3$ | Triangle | $ 8$ | Octagon | |
| $ 4$ | Quadrilateral | $ 9$ | Nonagon | |
| $ 5$ | Pentagon | $ 10$ | Decagon | |
| $ 6$ | Hexagon | $ 11$ | Hendecagon | |
| $ 7$ | Heptagon | $ 12$ | Dodecagon |
Angles of Polygons
The two theorems having to do the sums of angles of polygons:
- The Exterior Angle Sum states that the sum of the exterior angles of an $ n$-sided convex polygon is always $360{}^circ $. Thus, each exterior angle of the polygon measures $ displaystyle S=frac{{360}}{n}$, where $ n$ is the number of sides.
- The Interior Angle Sum states that the sum of the interior angles of an $ n$-sided convex polygon, one angle at each vertex, is $ displaystyle S=left( {n-2} right)times 180$, where $ n$ is the number of sides (weird, but this is because an $ n$-gon polygon can be divided up into $ n-2$ triangles, each with angles totaling $ 180{}^circ$). Thus, each interior angle of the polygon measures $ displaystyle S=frac{{left( {n-2} right)times 180}}{n}$. $
This is text let’s try a formula $ n-1$ $latex \displaystyle S=\frac{{360}}{n}$ .