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# Apothem

Apothem usually refers to a line segement in a regular polygon which connects the centre of the polygon to the centre of one of the sides of a polygon. You can also define it as the line drawn from the centre of the polygon perpendicular to one of the sides of the polygon. Only regular polygons are supposed to have Apothems. All Apothems are congruent in nature, meaning they are all equal in length.

Examples of Apothem: ## Apothem Definition

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Apothem is a line segment from the center of a regular polygon to the midpoint of a side. Its also defined as radius of the incircle of the polygon. For a regular polygon of n sides, there are n possible apothems of same length. Irregular Polygons have no apothem since they have no center.

## Properties of an Apothem

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Given below are some properties for an apothem:
• An Apothem always divides a n-sided regular polygon in to n isosceles triangles whose base is the side of the poygon and height being the apothem of the poygon.
• An apothem always represents the radius of a circle inscribed in the n-sided regular poygon.
• An apothem bisects the central angle determined by the side to which it is drawn.

## Apothem Formula

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Given below are the formulas for apothem:

Formula for the Apothem (a) = $\frac{s}{2 \tan \theta}$

Where, s is the side length and $\theta$ = $\frac{360^o}{2n}$

Alternate formulas for apothem length:

If radius of the polygon is given, then

Apothem = r cos ($\frac{180^o}{n}$)

Where, n is the number of sides.
If area of the polygon is given, then

Apothem = $\frac{2A}{P}$

Where, A is the area and P is the perimeter of the polygon.

## Derivation of Area of Polygon Using Apothem

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If n is the number of sides of a polygon and s is the length of each side of the polygon, then each apothem divides the n sided polygon in to n congruent isoseles triangles where the apothem forms the height of the triangle while the respective sides form the base of the triangle.

So, applying the formulae of area of triangel to a single isoseles traingle,

Area of the triangle = ($\frac{1}{2}$) × base × height

= $\frac{sa}{2}$

Since we have the area of a single traingle formed by the apothem, we just need to multiply it with the number of sides of the poygon to arrive at the area of the polygon. If we assume a n-sided polygon,

Area of the n-sided poygon An = $\frac{nsa}{2}$

If we represent the perimeter of the polygon to be P = ns, Area of the poygon = $\frac{Pa}{2}$

## Finding the Apothem

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If we assume a right angled triangle where the apothem is the adjacent side, half the side of the polygon to be the opposite side and line joining the centre of the polygon and vertex of the polygon to be the hypotenuse.

Lets assume the angle formed by adjacent side and hypotenuse to be $\theta$. Tan $\theta$ = $\frac{Opposite\ side}{Adjacent\ side}$

Tan $\theta$ = $\frac{\frac{s}{2}}{a}$ = $\frac{s}{2a}$

a = $\frac{s}{2 \tan \theta}$

The value of $\theta$ depending upon the number of sides of the polygon.

## Apothem of Hexagon

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The number of sides of the Hexagon is n = 6, and $\theta$ = $\frac{360^o}{2n}$ = $\frac{360^o}{12}$ = 30$^o$

Formula for the Apothem = $\frac{s}{2 \tan \theta}$

Where, s is the side length. => Apothem of Hexagon = $\frac{s}{2 \tan 30^o}$ = 0.86 s (Tan 30$^o$ = 0.58 ).

Apothem of the hexagon can also be found with the help of formula $\frac{2A}{P}$.

Let us find the apothem of a regular hexagon with side 10 feet and area 54 square feet.

Here, n = 6 and s = 10 feet and area (A) = 54 square feet

Apothem of regular hexagon = $\frac{2A}{P}$
where, A = area and P = perimeter of the polygon

Perimeter = 6 $\times$ n = 6 $\times$ 6 = 36

=> Apothem = $\frac{2. 54}{36}$

= 3

Apothem of the regular hexagon is 3 feet.

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