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.

## Definition

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

## Finding Area of a Polygon using 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 = $\frac{1}{2}$ $\times\ base\ \times\ height$ = $\frac{1}{2}$ $\times\ side\ of\ polygon\ \times\ apothem$

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

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

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.

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

## Apothem of a Triangle

If a triangle is a regular polygon then it must be an equilateral triangle Ã¢â‚¬â€œ 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 Square

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 $4 cm$.

## Examples

Example 1:

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

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.

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.

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.

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. Solution: Area =$\frac{1}{2}\times\ apothem\ \times\ perimeter66.03$=$\frac{1}{2}\times\ 4.26\ \times\ perimeter$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 =$(s/2)\ \times\ \sqrt 3$Area of polygon =$\frac{1}{2}\times\ n\ \times\ side\ of\ polygon\ \times\ apothem54 \sqrt 3$=$\frac{1}{2}\times\ 6\ \times\ s\ \times\ (s/2)\ \times\ \sqrt 354$=$6 /4\times\ s^{2}s^2$=$36s$=$6Perimeter$=$6\ \times\ 6$=$36 cm\$