In elementary geometry, a *polygon* is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or *circuit*. Or simply a closed plane figure bounded by three or more line segment to form a closed loop is called a polygon.

__Nomenclature:__

- The line segments forming the polygon are called sides.
- The point of intersection of two line segments is called a vertex.
- Number of vertices of a polygon is equal to the number of line segments or sides.

__Different types of Polygons: __ This is based on the number of sides that the polygon has. Here are few examples:

__Diagonal of a Polygon__: A line segment joining any two non-consecutive vertices is called

a diagonal of the polygon.

The dotted lines are diagonals of the shown polygons.

Interior and Exterior Angles of a Polygon: This is an important concept.

are interior angles. These are made by the two sides of the polygon.

is called an exterior angle. This is formed by extending a side of the polygon as shown in the adjacent figure.

Just by looking at the figure you can tell that

Hence we can say that:

Exterior Angle + Adjacent Interior Angle

__Convex Polygon vs Concave Polygon__

If the interior angle of the polygon is less than , then it is called convex polygon. If you look at any of the polygons shown above, you will see that all the interior angles are less than .

But there can be cases where the interior angle of a polygon could be more than . Take a look at the adjacent figure. Here you will see that (which is a reflex angle).

__Regular Polygon:__ A polygon that satisfies the following condition is called a regular polygon.

- All sides are equal
- All interior angles are equal
- All exterior angles are equal

For a regular polygon with $latex n &s=1$ sides we have the following:

Each Interior Angle

Proof:

A polygon can be divided into triangles.

See a few examples in the adjacent figure.

We know that the sum of the angles of a triangle is .

Therefore the sum of the interior angles

Proof:

Sum of all Interior Angles + Sum of all Exterior Angles

Sum of all Exterior Angles

3. From the above point 2, we can also say that

4. We already know that

Exterior Angle + Adjacent Interior Angle

Hi

The more intersting thing is not the exterior angels but the center angels where you have their sum of 360 ° .

A single center angel therefor has 360°/n .

same 2×pi/n

How can you count the hole circle in U=2×pi×r in the infinity

If you think of this you construkt a spiral of U=squr of 2 × pi

U (center)=U or U (interior)=U

exterior is trivial if you think of a quader

Good luck

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