Apothem
The apothem of a regular polygon is the line segment that connects the center of the polygon to the midpoint of one of its sides and is perpendicular to that side.
In practical terms, the apothem is simply the shortest distance from the center of a regular polygon to one of its sides.
The apothem plays an important role in geometry because it can be used to calculate several properties of regular polygons, including their area.
In every regular polygon, the apothem is also the radius of the inscribed circle, commonly called the inradius.
The Relationship Between the Apothem and the Side Length
For a given regular polygon, the ratio between the apothem \(a\) and the side length \(l\) is always the same. This ratio is known as the polygon's apothem constant, denoted by \(f\).
$$ f = \frac{a}{l} $$
The value of \(f\) depends only on the number of sides of the polygon.
Whether the polygon is large or small, the ratio remains unchanged.
This means that every type of regular polygon has its own characteristic apothem constant.
For example, the apothem constant of a regular pentagon is approximately 0.688, while that of a regular hexagon is approximately 0.866. Since these values are irrational numbers, they are usually rounded when performing calculations.
The following table lists the approximate values of the apothem constant \(f\) for some common regular polygons:
| Regular Polygon | Apothem Constant \(f\) |
|---|---|
| Equilateral Triangle | 0.289... |
| Square | 0.5 |
| Regular Pentagon | 0.688... |
| Regular Hexagon | 0.866... |
| Regular Heptagon | 1.038... |
| Regular Octagon | 1.207... |
| Regular Nonagon | 1.374... |
| Regular Decagon | 1.539... |
| Regular Dodecagon | 1.866... |
Using the Apothem Constant
The apothem constant provides a quick way to move between the side length and the apothem.
If you know the apothem, you can find the side length:
$$ l = \frac{a}{f} $$
If you know the side length, you can find the apothem:
$$ a = l \cdot f $$
Consider a regular hexagon with side length 6 cm.
Since the apothem constant of a regular hexagon is approximately \(f = 0.866\), the apothem is:
$$ a = l \cdot f $$
$$ a = 6 \cdot 0.866 $$
$$ a = 5.196 $$
Therefore, the apothem of the hexagon is approximately 5.196 cm.
Calculating the Area of a Regular Polygon
The apothem constant becomes especially useful when calculating the area of a regular polygon.
The standard area formula is:
$$ A = n \cdot \frac{l \cdot a}{2} $$
where:
- \(A\) is the area of the polygon
- \(n\) is the number of sides
- \(l\) is the side length
- \(a\) is the apothem
Since \(a = l \cdot f\), we can substitute the apothem into the formula:
$$ A = n \cdot \frac{l \cdot l \cdot f}{2} $$
$$ A = n \cdot \frac{l^2 \cdot f}{2} $$
At this point, it is convenient to introduce another constant:
$$ \phi = \frac{n \cdot f}{2} $$
This value depends only on the type of regular polygon and allows the area formula to be written in a simpler form:
$$ A = l^2 \cdot \phi $$
The following table shows some approximate values of the \(\phi\) constant:
| Regular Polygon | Constant \(\phi\) |
|---|---|
| Equilateral Triangle | 0.433... |
| Square | 1 |
| Regular Pentagon | 1.72... |
| Regular Hexagon | 2.598... |
| Regular Heptagon | 3.634... |
| Regular Octagon | 4.828... |
| Regular Nonagon | 6.182... |
| Regular Decagon | 7.694... |
| Regular Dodecagon | 11.196... |
Finding the Side Length from the Area
Note: The \(\phi\) constant is also useful when working backward from the area.
If the area of a regular polygon is known, the side length can be calculated using:
$$ l = \sqrt{\frac{A}{\phi}} $$
This formula is obtained by rearranging the area equation:
$$ A = l^2 \cdot \phi $$
$$ l^2 = \frac{A}{\phi} $$
$$ \sqrt{l^2} = \sqrt{\frac{A}{\phi}} $$
$$ l = \sqrt{\frac{A}{\phi}} $$
Similar inverse formulas can be derived to calculate any of the other quantities involved.
