The apothem of a hexagon is the length of the segment that connects the center of the hexagon to the midpoint of one of its sides. This segment is perpendicular to the side and plays an important role in geometry. The apothem is especially useful when calculating the area of a hexagon. Knowing it makes the computations easier and allows you to use more convenient formulas.
One common way to find the apothem is to divide the hexagon into six equal triangles and use one of those triangles to compute the apothem. In this case, you can apply the Pythagorean theorem or trigonometric methods to obtain formulas that help you determine this value accurately.
Apothem of a Hexagon: From the Pythagorean Theorem to Trigonometry
There are two main methods you can use to derive a formula for the apothem of a hexagon. Both methods rely on splitting the hexagon into six congruent triangles, as shown in the image below:
Using the Pythagorean Theorem
One way to find the apothem of a hexagon is to use the Pythagorean theorem on one of the triangles formed. We know that each of these triangles is equilateral, so all three sides are equal. If we draw the apothem, we can see that it splits the base into two equal parts, as shown in the figure:
So, using these lengths, we have:
\[ AO^2 = AG^2 + OG^2. \]
Further simplification gives the formula for finding \( OG \) (the apothem of a hexagon):
\[ \begin{array}{@{}l@{\qquad}l@{\qquad}l@{}} OG^2 = AO^2 – AG^2 = AO^2 – \left(\dfrac{AO}{2}\right)^2, & OG^2 = AO^2 – \dfrac{AO^2}{4} = \dfrac{3 \cdot AO^2}{4}, & OG = \dfrac{\sqrt{3} \cdot AO}{2}. \end{array} \]
Using Trigonometry
The second method also starts by dividing the hexagon into six equal triangles. When we draw the apothem, we split each of these triangles in half, which produces \( 12 \) smaller triangles.
For the trigonometric approach, we need the central angle of each small triangle. Since the full angle around the center is \( 360^\circ \), each small triangle has an angle of \( 30^\circ \) (\( \frac{360^\circ}{12} \)).
Now we can use the tangent function. The tangent of an angle equals the ratio of the opposite side to the adjacent side:
\[ \tan(30^\circ) = \frac{AG}{OG}. \]
From this, we solve for \( OG \):
\[ \begin{array}{@{}l@{\qquad}l@{}} \tan(30^\circ)=\dfrac{AG}{OG}=\dfrac{\dfrac{AO}{2}}{OG}, & OG=\dfrac{AO}{2 \cdot \tan(30^\circ)}. \end{array} \]
Note. If we denote the side length of a regular hexagon by \( a \) and the apothem by \( h \), then the formulas take their familiar form:
\[ \begin{array}{@{}l@{\qquad}l@{}} h=\dfrac{\sqrt{3} \cdot a}{2}, & h=\dfrac{a}{2 \cdot \tan(30^\circ)}. \end{array} \]
Practice Problems and Solutions: Apothem of a Hexagon in Examples
In the examples below, we use both apothem formulas written earlier. Each example includes a solution, but it’s a good idea to try solving the exercises on your own before checking the result.
Example 1. What is the apothem of a hexagon with side length 6 cm?
Using the first formula with \( a=6 \), we get:
\[ h=\frac{\sqrt{3} \cdot a}{2}=\frac{\sqrt{3} \cdot 6}{2}\approx 5.2. \]
So, the apothem of the hexagon is \( 5.2 \) cm. If we use the second formula with \( a=6 \) cm, we have:
\[ h=\frac{a}{2 \cdot \tan\left(30^\circ\right)}=\frac{6}{2 \cdot \tan\left(30^\circ\right)}\approx 5.2. \]
So, both formulas give the same result.
Example 2. A hexagon has side length 8 cm. What is the length of its apothem?
Let’s use the first formula with \( a=8 \):
\[ h=\frac{\sqrt{3} \cdot a}{2}=\frac{\sqrt{3} \cdot 8}{2}\approx 6.93. \]
So, the apothem of the hexagon is \( 6.93 \) cm. Using the second formula with \( a=8 \) cm, we get:
\[ h=\frac{a}{2 \cdot \tan\left(30^\circ\right)}=\frac{8}{2 \cdot \tan\left(30^\circ\right)}\approx 6.93. \]
So, just like in the previous example, the result is the same.
Example 3. What is the side length of a hexagon with an apothem of 12 cm?
Here, the apothem is known, and we need to find the side length. Using the first formula with \( h=12 \), we solve for \( a \):
\[ \begin{array}{llll} h=\dfrac{\sqrt{3} \cdot a}{2}, & 12=\dfrac{\sqrt{3} \cdot a}{2}, & 24=\sqrt{3} \cdot a, & a=\dfrac{24}{\sqrt{3}}=8 \cdot \sqrt{3}\approx 13.86. \end{array} \]
So, the side length is about \( 13.86 \) cm. If we use the trigonometric formula with \( h=12 \) cm, we get:
\[ \begin{array}{llll} h=\dfrac{a}{2\cdot \tan(30^\circ)}, & 12=\dfrac{a}{2\cdot \tan(30^\circ)}, & a=12\cdot\bigl(2\cdot \tan(30^\circ)\bigr)\approx 13.86. \end{array} \]
So, both formulas give the same result: \( a\approx 13.86 \) cm.
See Also: What Else Is Worth Learning About Hexagons
If you already feel confident with the apothem of a hexagon, it’s a good moment to take the next step. Below are topics that will help you understand hexagons better and solve problems more confidently:
- Diagonals of a Hexagon: Explanation, Properties, and Examples — You’ll learn how many diagonals a hexagon has and how to count them quickly in problem-solving.
- Perimeter of a Hexagon: Calculation and Practical Uses — You’ll practice finding the perimeter for different types of hexagons and see how this is used in real examples.
- Area of a Hexagon: Formulas and Worked Examples — You’ll explore several ways to calculate area and understand where the apothem gives the biggest benefit.
From Formula to Code: Implement the Flowchart in a Program
If you’re interested not only in understanding what the apothem of a hexagon is, but also in using this idea in practice, try writing a small app in your favorite programming language. This kind of task clearly shows how a mathematical formula turns into a useful tool: the program checks the input, calculates the apothem, and displays the result in a convenient format.
A flowchart can really help here because it gives you a clear, step-by-step logic that’s easy to translate into Python, JavaScript, C#, Java, or any other language you enjoy working with.
