** **A **polygon** is a two-dimensional object; it is a plane shape with straight sides. A regular polygon is defined as a polygon which is all the sides are
congruent and all the sides of interior angles are congruent; that is a polygon, both equiangular and equilateral. Important point is all the regular polygons are convex. The word
"**apothem"** is referred as the length of that line segment.

Apothem is defined as which is a line segment from the center of the regular polygon to the midpoint of one of its sides. Otherwise apothem is defined as the perpendicular line or distance from the center of a regular polygon to any of its sides.

**Example for apothem: Hexagon**

**Formula for apothem:**

Apothem = the sides of the polygon is divided by two times (twice) the tangent of the half of the midpoint (central) angle of the polygon

Commonly it is the line drawn from center of the polygon that is perpendicular to one of all sides.

**Regular Polygon:**

** **1. A polygon is a 2-dimensional object; it is a plane shape with straight sides.

2. Commonly A **"Regular Polygon"** has included the following two conditions:
All **sides** are equal, and all **angles** are equal.

3. Examples for regular polygons: Like that triangles, squares (quadrilateral), pentagons, hexagons and so on...

Example: Pentagon (Regular polygon).

4. Regular polygons are only polygons that having apothems, all the apothems in a polygon will be congruent and have the same length of all the sides. So that its only having apothem.

The apothem **‘a’** can be used to find the area of any regular n-sided polygon of side
length **‘s’** according to the following formula, which also states that the area is equal to the apothem multiplied
by half the perimeter since **ns = p.**

** A = nsa / 2**

** = pa / 2**

This formula will be derived by partitioning the n-sided polygon into **n** congruent isosceles triangles, and noting that the apothem is the height of each triangle, and that area
of the triangle equals half of the base times to the height.

An apothem of a regular polygon is being a radius of inscribed circle. So that is also small distance from any side of the polygon to its center.

1. first way of finding apothem:

The apothem of a regular polygon can be found in so many ways, which two are described here:

The apothem of a regular n-sided polygon with side length S, or circum radius,

So it can be find used this following formula:

a = s / 2 tan ( / n )

= R cos (180 /n ).

2. Another way to find the apothemuse this formula:

a = 1/2 s tan ( 90 (n - 2) / n )

Here the two formulas can still be used, because of even if only the perimeter P and the number of sides n are known.

s = p / n.