Polygons

Apothem of a Pentagon: How to Find It from the Side

The apothem of a pentagon is the perpendicular distance from the center of a regular pentagon to one of its sides (that is, to the point where the inscribed circle touches the side). You can also think of it as the radius of the circle inscribed in a regular pentagon. Most often, the apothem is used when you need to calculate the area of a regular polygon.

In this article, we’ll learn how to find the apothem of a pentagon when the side length is known. And of course, we’ll also apply the formula in a few practical examples.

Step by Step to the Formula: Apothem of a Pentagon

To derive the apothem formula, let’s look at a regular pentagon \( ABCDE \) with center \( O \). Draw segments from the center to all vertices—this splits the pentagon into five equal isosceles triangles. Take one of them, for example triangle \( AOB \).

Pentagon ABCDE, divided into five equal triangles from the center, showing the apothem as a perpendicular to a side

Let \( F \) be the midpoint of side \( AB \). Then the segment \( OF \) is perpendicular to \( AB \) and is the apothem. At the same time, \( OF \) is the height of triangle \( AOB \), and it bisects the base \( AB \), so we have:

\[ AF = FB = \frac{AB}{2}. \]

Now let’s look at the central angle. A full angle around point \( O \) is \( 360^\circ \). Since there are five triangles, we get:

\[ \angle AOB = \frac{360^\circ}{5} = 72^\circ. \]

The height \( OF \) divides triangle \( AOB \) into two equal right triangles, so:

\[ \angle AOF = \frac{72^\circ}{2} = 36^\circ. \]

Now consider the right triangle \( AOF \). In it:

  • The leg opposite the angle \( 36^\circ \) is \( AF=\dfrac{AB}{2} \).
  • The adjacent leg is \( OF \) (the apothem).

In the right triangl AOF, OF is perpendicular to AB and represents the apothem

By the definition of tangent:

\[ \tan(36^\circ)=\frac{AF}{OF}=\frac{\dfrac{AB}{2}}{OF}. \]

From this, we get:

\[ OF=\frac{AB}{2\cdot\tan(36^\circ)}. \]

Note. If we denote the side length of the regular pentagon by \( a \) and the apothem by \( h \), then the formula takes the familiar form:

\[ h=\dfrac{a}{2 \cdot \tan(36^\circ)}. \]

Apothem of a Pentagon: Examples with Answers

The apothem of a pentagon formula is used to solve the following examples. Each example comes with a solution, but we recommend trying to solve the problem on your own first before checking the answer.

Example 1. What is the apothem of a pentagon with side 4 cm?

We are given \( a=4 \) cm. Using the apothem formula, we get:

\[ h=\frac{a}{2\cdot\tan\left(36^\circ\right)}=\frac{4}{2\cdot\tan\left(36^\circ\right)}\approx 2.75. \]

So, the apothem of the pentagon is \( 2.75 \) cm.

Example 2. A pentagon has a side length of 5 cm. What is its apothem?

In this case, \( a=5 \) cm. Substitute into the formula:

\[ h=\frac{a}{2\cdot \tan\left(36^\circ\right)}=\frac{5}{2\cdot\tan\left(36^\circ\right)}\approx 3.44. \]

So, the apothem of the pentagon is \( 3.44 \) cm.

Example 3. A pentagon has a side length of 10 cm. What is its apothem?

We are given \( a=10 \) cm. Substitute into \( h=\dfrac{a}{2\cdot\tan(36^\circ)} \):

\[ h=\frac{a}{2\cdot \tan\left(36^\circ\right)}=\frac{10}{2\cdot\tan\left(36^\circ\right)}\approx 6.88. \]

So, the apothem of the pentagon is \( 6.88 \) cm.

Example 4. The apothem of a pentagon is 7.6 cm. What is the side length?

Here we know \( h=7.6 \) cm. From \( h=\dfrac{a}{2\cdot\tan(36^\circ)} \), we find \( a \):

\[ h=\dfrac{a}{2\cdot \tan(36^\circ)},\qquad 7.6=\dfrac{a}{2\cdot\tan(36^\circ)},\qquad a=7.6\cdot\bigl(2\cdot\tan(36^\circ)\bigr)\approx 11.04. \]

So, the side length of the pentagon is \( 11.04 \) cm.

Example 5. What is the side length of a pentagon with apothem 6 cm?

We have \( h=6 \) cm. Let’s find \( a \):

\[ h=\dfrac{a}{2\cdot \tan(36^\circ)},\qquad 6=\dfrac{a}{2\cdot \tan(36^\circ)},\qquad a=6\cdot\bigl(2\cdot \tan(36^\circ)\bigr)\approx 8.72. \]

So, the side length of the pentagon is \( 8.72 \) cm.

See Also: What Else Is Worth Reviewing About a Pentagon

Want to understand pentagons better and do calculations faster? Then these materials will be a great fit:

  1. Perimeter of a Pentagon: Formulas and Examples — A short, step-by-step guide on how to find the perimeter of a pentagon without getting confused by formulas and substitutions.
  2. Area of a Pentagon: Formulas and Examples — We explain how to calculate the area of a regular pentagon using the side length and the apothem, with clear examples.
  3. Interior Angles of a Polygon: Formula and Examples — You’ll learn how to find the sum and the measure of angles in a pentagon and how to use that in problem-solving.

From Formula to Code: Want to Program the Apothem?

And now—something especially fun for those who enjoy programming. Take a look at the flowchart below and try writing a program that does the same thing, but in your favorite language. It could be Python, JavaScript, C#, Java, or even PHP for a website.

A nice bonus is that you can quickly turn the result into a small “geometry calculator”. Handy, right? And when your code outputs the apothem of a pentagon from the side length you enter, you’ll really feel that the formula works not only on paper.

A flowchart of the algorithm showing how the apothem of a pentagon is calculated from the side length