# Apothem

Apothem

Tim and Sam were finding the area of various regular polygons. Tim divided the polygons into triangles and was trying to calculate the area of the polygons, which took him too long. On the other hand, Sam found the area of the polygons easily, as he knew the length of the apothem. So let's learn about apothem today!

Before we get started, check out this interesting simulation to identify the apothem for various polygons. This apothem calculator will help you understand the lesson better.

## Lesson Plan

 1 What Is Meant by Apothem? 2 Important Notes on Apothem 3 Solved Examples on Apothem 4 Challenging Questions on Apothem 5 Interactive Questions on Apothem

## What Is Meant by Apothem?

Apothem is a line drawn from the center of any polygon to the midpoint of one of the sides. ## Formulas Used to Calculate the Apothem Length

The apothem formula, when the side length is given is:

 $$a$$ = \begin{align}\frac{S}{2\,\, \text{tan}\left ( \frac{180}{n} \right )}\end{align}

Where,

$$a$$ = apothem length

$$s$$ = side length

$$n$$ = number of sides of a polygon.

The apothem formula , when the radius is given is:

 $$a$$ = $$r.cos\frac{180}{n}$$
Where

$$r$$ = radius.

$$n$$ = number of sides

$$Cos$$ = cosine function which is calculated in degrees.

We can use the apothem area formula of a polygon to calculate the length of the apothem.

 $$A$$ = $$\dfrac{1}{2}aP$$

Where,

$$A$$ = area of the polygon

$$a$$ = apothem.

$$P$$ = perimeter

## How to Calculate Area of a Polygon Using Apothem?

To calculate the area of a polygon with the help of apothem, we use the formula:

$$A$$ = $$\dfrac{1}{2}aP$$

Where,

$$a$$ = apothem.

$$P$$ = perimeter.

Example: Find the area of a regular hexagon, if the side length is $$5$$ inches, and the apothem is $$3$$ inches.

$$A$$ = $$\dfrac{1}{2}aP$$

As we know perimeter:

\begin{align}P &= [\text{side length}]\times [\text{no. of sides}]\\&= 5\times6 = 30\end{align}

After the perimeter is calculated, we use it in the formula of Area = $$A$$ = $$\dfrac{1}{2}aP$$

\begin{align}A &= \dfrac{1}{2}aP \\ & = \dfrac{1}{2} \left ( 3\right )\left (30 \right )\\& = 45 \text{ inches}^2\end{align}

More Important Topics
Numbers
Algebra
Geometry
Measurement
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Trigonometry
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More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus