Apothem is a line joining the center of the polygon to one of its sides. We can develop a convenient formula for finding the apothem of a regular polygon. Since an n-sided regular polygon is composed of n identical triangles, the formula for the apothem of a regular polygon is found by dividing 2 times the area of polygon by the perimeter of the regular polygon. 
Properties :
1) An apothem of a regular polygon is the perpendicular bisector of the side to which it is drawn.

2) An apothem bisects the central angle determined by the side to which it is drawn.

Apothem is the line from center of a regular polygon to midpoint of a side. Apothem is perpendicular bisector of the side. In a polygon of n sides there are $n$ possible apothems. 

Since irregular polygons have no center they have no apothems. The word apothem can refer to the line itself and also to its length.

$ABCDEF$ is a regular hexagon and $GH$ is apothem from center $G$ to $H$ the midpoint of side $BC$.

$<\ GHB$ is $90^{\circ}$.

Apothem Definition


Apothem of a polygon is its incenter circle with apothem as radius which touches every side of the polygon.

Definition of Apothem

To use apothem for finding area we have to visualize a polygon of n sides as being made up of n number of triangles. Each side of polygon is the base of triangle and apothem makes up the height of triangle.

Area of triangle is = $\frac{1}{2}$ $\times\ base\ \times\ height$ = $\frac{1}{2}$ $\times\ side\ of\ polygon\ \times\ apothem$

Area of Polygon using Apothem

To find the area of the entire polygon, we find sum of the area of n triangles. 

Area of polygon = $\frac{1}{2}$ $\times\ n\ \times\ side\ of\ polygon\ \times\ apothem$  = $\frac{1}{2}$ $\times\ perimeter\ \times\ apothem$

Area of Polygon Example

Example 1:

Let the apothem be $4.33 cm$ and each side of he\timesagon be $5 cm$

Solution:

Area of polygon = $\frac{1}{2}$ $\times\ perimeter\ \times\ apothem$
                
          =  $\frac{1}{2}$ $\times\ n\ \times\ side\ of\ polygon\ \times\ apothem$

  = $\frac{1}{2}$ $\times\ 4.33\ \times\ 5\ \times\ 6$ = $64.95\ square\ cm$


If apothem is not given but only side is given we can find apothem using tan ratio from trigonometry.

We split one of the triangles in half. The apex angle is $\frac{360}{2n}$ shown by angle $x$ in the diagram.

Apothem of a Polygon

$Tan\ x$ = $\frac{side/2}{apothem}$
Example 2:

Let the side of a regular heptagon be $7 cm$. Find its area.

Solution:

We first find apex angle using formula $\frac{360}{2n}$ where $n$ is number of sides. 

Apex angle $\frac{360}{7}$ (2) = $25.71^{\circ}$

To find apothem we use the formula stated above 

$Tan\ x$ = $\frac{side/2}{apothem}$

$Tan\ 25.71$ = $\frac{7/2}{apothem}$

$0.4814$ = $\frac{3.5}{apothem}$

Apothem = $7.27 cm$

Area of polygon = $\frac{1}{2}$ $\times\ perimeter\ \times\ apothem$
                
  = $\frac{1}{2}$ $\times\ n\ \times\ side\ of\ polygon\ \times\ apothem$

  = $\frac{1}{2}$ $\times\ 7.27\ \times\ 7\ \times\ 7$ = $178.11$ square cm
If a triangle is a regular polygon then it must be an equilateral triangle – all sides are equal and each angle $60^{\circ}$

The length of apothem of an equilateral triangle is the short leg of the $30 - 60 - 90$ triangle where $s/2$ is the long leg.

In the diagram apothem $i$ = $\frac{(s/2)}{\sqrt 3}$ = $\frac{(s/2) \sqrt 3}{3}$

Apothem of a Triangle
The length of apothem of a square is half the length of a side.

Length of apothem $EF$ of square $ABCD$ is $2 cm$ since side length is $4 cm$.

Apothem of a Square
Example 1: 

In an equilateral triangle, the apothem is $\frac{4 \sqrt 3}{3}$ inches and each side is $8$ inches. Find area.

Apothem Example

Solution:

Area = $\frac{1}{2}$ $\times$ $\frac{4 \sqrt 3}{3}$ $\times\ 8\ \times\ 3$ = $16 \sqrt 3$ square inches
Example 2:

In a regular hexagon, each side is $4$ ft and length of apothem is $3.46$ ft. Find area.

Example on Apothem

Solution: 

Area = $\frac{1}{2}$ $\times\ 3.46\ \times\ 4\ \times\ 6$ = $41.52$ square ft

Example 3:

In a regular octagon each side is $11$ inches long and length of apothem is $13.27$ inches. Find area.

How to find Apothem

Solution:

Area = $\frac{1}{2}$ $\times 13.27\ \times\ 8\ \times\ 11$ = $583.88$ square inches 
Example 4:

Find the perimeter of a regular hexagon that has an area of $392.85$ square cm and apothem measuring $10.65$ cm.

Problem on Apothem

Solution:

Area = $\frac{1}{2}$ $\times\ apothem\ \times\ perimeter$

$392.85$ = $\frac{1}{2}$ \times\ 10.65\ \times\ perimeter$

Perimeter = $392.85 $ $(\frac{1}{2}$ $\times\ 10.65)$ = $73.77 cm$
Example 5:

In a regular pentagon the area is $66.03$ square ft and apothem is $4.26$ ft. Find length of each side.

Find Length of Apothem

Solution:

Area = $\frac{1}{2}$ $\times\ apothem\ \times\ perimeter$

$66.03$ = $\frac{1}{2}$ $\times\ 4.26\ \times\ perimeter$

Perimeter = $66.03  $ $(\frac{1}{2}$ $\times\ 4.26)$ = $31 ft$

Length of each side = $31 *  5$ = $6.2 ft$
Example 6:

Find the perimeter of a regular hexagon that has an area of $54 \sqrt 3$ square cm

Solution:

A regular hexagon is made up of $6$ equilateral triangles of side $s$ each, which can be split into $2$ right triangles with $30 - 60 - 90$ angles. 

The $60^{\circ}$ is opposite to the apothem and $30^{\circ}$ is opposite to $s/2$ as shown in diagram

Apothem of a Regular Polygon

Apothem = $(s/2)\ \times\ \sqrt 3$

Area of polygon = $\frac{1}{2}$ $\times\ n\ \times\ side\ of\ polygon\ \times\ apothem$

$54 \sqrt 3$ = $\frac{1}{2}$ $\times\ 6\ \times\ s\ \times\ (s/2)\ \times\ \sqrt 3$

$54$ = $6 /4\times\ s^{2}$

$s^2$ = $36$

$s$ = $6$

$Perimeter$ = $6\ \times\ 6$ = $36 cm$