Contents

- Definition
- Finding a Polygon’s Apothem: The Step-by-Step Guide
- What Is Apothem Used For?
- How To Calculate Apothem Length
- What Exactly Is the Distinction Between an Apothem and a Radius?
- What Accounts for the Fact That the Radius Is Larger Than the Apothem?
- Apothemâ€™s Visual Portrayal
- Numerical Examples for Apothem

# Apothem|Definition & Meaning

## Definition

In a **regular** polygon, the **apothem** is the **segment** of a **line** that runs from the **center** to the **middle** point of a side. In other **words, it’s** the line that cuts through the **polygon** at **right angles** to one of the sides, **starting** at the **center.**

To **measure** the **apothem** of a **regular** polygon, find the **midpoint** of one of its sides and **measure** it from the center of the polygon. A segment as well as a unit of **measurement** in one. Regular **polygon** area **calculation** is the most common application of this method. The **sides** of a **regular** polygon are all the **same length,** and the polygon itself is **closed.** In regular polygons with **congruent** sides, **apothems** do not exist.Â

The following **figure represents** the apothem of **different** polygons.

The **apothem** functions as both the **center** and the radius of an incircle that **constitutes** the polygon. There are n **different** apothems for a polygon that has n **sides,** with each side **having** an **equal length** of the course. The term **“apothem”** can be used to refer to either the line itself or to the **distance** that line covers.

**Therefore,** it is possible for you to **appropriately** say “draw the apothem” and “the **apothem** is 4 **centimeters.”**

## Finding a Polygon’s Apothem: The Step-by-Step Guide

In the process of **estimating** the area and **perimeters** of polygons, apothems can be of **great** use. **Finding** the apothem of a **polygon** requires **sufficient information,** which may be obtained by knowing its **perimeter** and area. The procedure for **determining** a **polygon’s apothem** is shown below.

To **begin,** let’s look at the **formula** for **calculating** the **area** of a **polygon:**

A = 0.5 x a x p

**Multiplying** “2” on **both** sides **results** in:

2 x A = a x p

**Dividing** “p” on both sides **results** in:

(2A)/p = a

- Where the
**polygon**area is represented by A. - The polygon
**perimeter**is represented by P. - “a” represents the
**apothem.**

## What Is Apothem Used For?

According to the following formula, the **apothem** a can be used to determine the **area** of any regular **n-sided polygon** with side length s. The formula **furthermore** states that the **area** is **equivalent** to the apothem calculated by **multiplying** by half the **perimeter** because **ns = p**. You can use this **information** to calculate the **area** of any regular **n-sided polygon.** A normal polygon will always have an **apothem** that is a radius of the **circular circle.**

## How To Calculate Apothem Length

There are **situations** where a **regular polygon’s** area is **unknown** and just its **side** lengths are known. **Another** apothem **formula** could be **applied** here:

apothem = $\displaystyle\frac{s}{2\tan \left( \frac{180}{n} \right)}$

- The
**regular polygon’s**side length is**denoted**by the letter s. - While n
**represents**the**number**of sides of the**polygon.**

## What Exactly Is the Distinction Between an Apothem and a Radius?

In a **regular** polygon, the radius **travels** from the **center** of the **shape** to each vertex, **where** it cuts the angle formed by the **vertex** in **half.** The **apothem** extends in a **straight** line from the **central** point to the **middle** of each side. The **side** is always cut in **half** by it, and it always runs **perpendicular** to the side.

## What Accounts for the Fact That the Radius Is Larger Than the Apothem?

In a regular polygon, the **apothem** is the **larger** leg of the right triangles created from the **isosceles** triangles, and the **hypotenuse** is the part of the triangle that corresponds to the radius. As a **consequence** of this, the **radius** is larger than the **apothem.**

## Apothemâ€™s Visual Portrayal

The **graphic** below **depicts** a visual **representation** of a **polygon** with **seven** sides and shows its **apothem.**

Figure 3 shows two **polygons** of **different** sides which **displays** the visual **representation** of **apothem.**

## Numerical Examples for Apothem

### Example 1

**Figure** out the **area** of a **polygon** that has an **apothem** of equalÂ 5 feet and a **perimeter** that has a **value** of 12 **feet.**

### Solution

We **know that** :

A = 0.5 x a x p

- Where the
**polygon area**is represented by A. - The
**polygon perimeter**is**represented**by P. - “a”
**represents**the**apothem.**

**Given that:**

**Apothem** = **5 feet**

**Perimeter** = **12 feet**

By **putting values,** we get:

A = 0.5(5)(12)

A = 0.5 x 60

**A = 30 sq. ft.**

Thus the **area** of the **polygon** is **30** sq.Â ft.

### Example 2

**Figure** out the **area** of a **polygon** that has an **apothem** of **equal** 5 **feet** and a **perimeter** that has a **value** of **48 feet.**

### Solution

We **know that** :

A =0.5 x a x p

- Where the
**polygon**area is**represented**by A. - The
**polygon**perimeter is**represented**by P. - “a”
**represents**the**apothem.**

**Given that:**

**Apothem** = **5 feet**

**Perimeter** = **48 feet**

By **putting values,** we get:

A = 0.5(5)(48)

A = 0.5(240)

A = 120 sq. ft.

**Thus** the **area** of the **polygon** is **120** sq. ft.

### Example 3

**What’s** the **apothem** for a **regular polygon with** a 204 square unit **area** and a **75 unit perimeter?**

**Solution**

**Given that** :

**Area** of **polygon** = 204 **square** unit

**Perimeter** = 75 **units.**

We need to find the apothem. We **know** that:

A = 0.5 x a x p

**Where**the**polygon**area is**represented**by**A**.- The
**polygon perimeter**is**represented**by**P**. - “a”
**represents**the**apothem.**

**Rearranging** the **above formula** results in the following:

a = (2A)/p

By **putting values,** we get:

a = 2(204)/75

a = 408/75

a = 5.44

**Thus** the **apothem** for the **given data** is 5.44.

**Example 4**

**Think** about the heptagon in the figure, **whose** sides are **each** 4 **units** long. Can you tell me the **apothem** of a **polygon?**

**Solution**

**Given** that:

Size **length** = 4 **units**

No of **sides** = 7

We need to find the **length** of the **apothem. **We know that**:**

apothem = $\displaystyle\frac{s}{2\tan\left( \frac{180}{n} \right)}$

- The regular
**polygon’s**side**length**is**denoted**by the**letter**s. - While n
**represents**the number of**sides**of the**polygon.**

By **putting** values, we get:

a = $\displaystyle\frac{4}{2\tan \left(\frac{180}{7}\right)}$

a= $\dfrac{4}{2\tan (25.71)}$

a = $\dfrac{4}{2(0.48)}$

a= $\dfrac{4}{2(0.48)}$

a = $\dfrac{4}{0.96}$

**a = 4.153**

Thus the **length** of **apothem** for the given **data** is **equal** to **4.153.**

*All images/mathematical drawings were created with GeoGebra.*