The perimeter of a pentagon is simply the sum of the lengths of all its sides.
A pentagon is a polygon with five sides, five vertices, and five interior angles whose total is \( 540^\circ \).
Pentagons can be regular or irregular, depending on whether all sides are the same length. Now let’s go through the main ways to calculate the perimeter of a pentagon.
Perimeter of a Pentagon: Formulas From Basic Sum to Trigonometry
The perimeter of a pentagon is the total length around the shape. So, for a pentagon \( ABCDE \) shown in the figure below, we write:
\[
P=AB+BC+CD+DE+AE,
\]
where \( P \) is the perimeter.
If the pentagon is regular, then all five sides are equal. In that case:
\[
P=5 \cdot AB.
\]
Note. If we label the side lengths as \( a \), \( b \), \( c \), \( d \), and \( e \), then the formulas can be written in a more familiar way:
\[
P = a + b + c + d + e,\qquad P = 5\cdot a.
\]
Perimeter of a Pentagon Using the Circumradius
You can find the perimeter of a pentagon if the radius of the circumscribed circle is given (this radius is also called the circumradius).
Important: this method applies only to a regular pentagon inscribed in a circle.
Here, the radius is the segment from the center \( O \) to a vertex. First, we find the side length:
\[
AB = 2\cdot AO\cdot \sin\left(\frac{180^\circ}{5}\right)=2\cdot AO\cdot \sin(36^\circ).
\]
Then we calculate the perimeter:
\[
P=5\cdot AB.
\]
Perimeter of a Pentagon Using the Apothem
You can also calculate the perimeter of a pentagon if the apothem is known.
Note: this method also applies only to a regular pentagon.
The apothem is a perpendicular segment drawn from the center of the pentagon to one of its sides (to the point where the inscribed circle touches the side). If the apothem is known, the side length can be found like this:
\[
AB = 2\cdot OF\cdot \tan\left(\frac{180^\circ}{5}\right)=2\cdot OF\cdot \tan(36^\circ).
\]
Once you have the side length, the perimeter is:
\[
P=5\cdot AB.
\]
Perimeter of a Pentagon: Examples with Answers
To better understand how to find the perimeter of a pentagon, let’s work through a few examples. Each one includes a complete solution. However, it’s a good idea to try the calculations on your own first—and then check your answer.
Example 1. Find the perimeter if the sides are 5 cm, 4 cm, 7 cm, 3 cm, and 6 cm
For an irregular pentagon, add all five side lengths:
\[
P=a+b+c+d+e=5+4+7+3+6=25.
\]
So, the perimeter is \( 25 \) cm.
Example 2. Find the perimeter of a regular pentagon with side length 10 cm
Since the pentagon is regular:
\[
P=5\cdot a=5\cdot 10=50.
\]
So, the perimeter is \( 50 \) cm.
Example 3. If the perimeter of a regular pentagon is 25 cm, find one side length
Use the perimeter formula for a regular pentagon:
\[
P=5\cdot a,\qquad 25=5\cdot a,\qquad a=5.
\]
So, the side length is \( 5 \) cm.
Example 4. Find the perimeter of a regular pentagon inscribed in a circle with radius 5 cm
First, calculate the side length:
\[
AB = 2\cdot AO\cdot \sin(36^\circ)=2\cdot 5\cdot 0.587785 \approx 5.88.
\]
Then the perimeter is:
\[
P=5\cdot AB \approx 5\cdot 5.88 \approx 29.39.
\]
So, the perimeter is approximately \( 29.39 \) cm.
Example 5. Find the perimeter of a regular pentagon if the apothem OF=5 cm
First, calculate the side length:
\[
AB=2\cdot OF\cdot \tan(36^\circ)=2\cdot 5\cdot 0.726543 \approx 7.27.
\]
Then the perimeter is:
\[
P=5\cdot AB \approx 5\cdot 7.27 \approx 36.33.
\]
So, the perimeter is approximately \( 36.33 \) cm.
See Also: Helpful Materials for Practice
Want to reinforce the topic and quickly expand your understanding of pentagons? Here’s a short list of resources that naturally complement this article:
- Apothem of a Pentagon: Formula and Examples — A clear, visual explanation of what an apothem is and how to find it in problems.
- Area of a Pentagon: Formulas and Examples — Shows how to calculate the area of a regular pentagon using the side length and the apothem, with worked examples.
- Interior Angles of a Polygon: Formula and Examples — Explains how to find the sum of interior angles and the measure of angles in a pentagon, and how to apply this in tasks.
Geometry in Code: Turn a Flowchart into a Program
If you enjoy not only calculating but also programming, try turning this flowchart into working code in your favorite language. It’s a small task, but it’s very practical. It helps you practice input handling, checks, branching (if/else logic), and cleanly displaying the final result. And yes—it’s satisfying to see how a geometry formula becomes part of a real program.
Write your own version, test it with different values, and you’ll quickly feel how the algorithm starts working in practice.
