# Apothem: Definition & Formula

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• 0:00 Definition of an Apothem
• 1:17 Apothem & Area
• 2:08 More Formulas
• 3:50 Example
• 5:04 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

An apothem is a part of a regular polygon. Through definitions, formulas, and examples, we will learn what an apothem is and how it can be used to analyze a regular polygon.

## Definition of an Apothem

In mathematics, a regular polygon is a polygon with n sides, all having equal length. Each regular polygon has a radius, an apothem, an incircle, and a circumcircle.

The radius of a regular polygon is the line connecting the center of the polygon to one of its vertices

The circumcircle is the circle around the outside of the polygon connecting all of its vertices. The radius of the polygon is also the radius of the circumcircle.

The incircle is the circle on the inside of the polygon touching each of the midpoints of the sides.

Lastly, the apothem is the line connecting the center of the polygon to the midpoint of one of the polygon's sides. The apothem is also the radius of the incircle.

You may be wondering what an apothem would look like on an irregular polygon, or a polygon with sides of different lengths. Irregular polygons have no center point, so they do not have an apothem. Makes sense, right? Apothems only apply to regular polygons.

Each one of the regular polygons shown has an apothem. A regular polygon of n sides always has an apothem no matter how many sides it has. The length of the apothem can be used to calculate other characteristics of a polygon.

## Apothem and Area

The perimeter of a regular polygon is the distance around the polygon. If we know the length of the apothem and perimeter of a regular polygon, we can calculate the area of the polygon using the formula:

A = (1/2)aP

where a is the length of the apothem and P is the perimeter.

For example, let's consider a 5-sided regular polygon with an apothem length of 4.817 units and a perimeter of 35 units. The area of this polygon can be found by plugging 4.817 in for a and 35 in for P in the formula to get:

A = (1/2)(4.817)(35) = 84.2975 square units

The area of this polygon is 84.2975 square units.

## More Formulas

We can also use the area formula to find the apothem if we know both the area and perimeter of a polygon. This is because we can solve for a in the formula, A = (1/2)aP, by multiplying both sides by 2 and dividing by P to get 2A / P = a.

Let's return to the image of the 5-sided regular polygon and assume that, while we don't know the apothem, we do know that the area is 84.2975 square units, and the perimeter is 35 units. When we plug these numbers into the apothem formula, we get:

a = 2(84.2975) / 35 = 4.817 units

Here, the apothem has a length of 4.817 units.

Sometimes, we don't know the area or perimeter of an n-sided regular polygon. When this is the case, we do have a formula we can use to find the length of the apothem knowing only the length of a side of a polygon. When we only know the length of a side, we call it s of an n-sided regular polygon, and we can use the following formula:

a = s / (2tan(180 / n))

to find the length of the apothem.

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