# Pentagon

The pentagon is a geometric figure made up of five sides, in addition to having five vertices and five internal angles.

That is, the pentagon is a polygon that has five sides, being of greater complexity than a quadrilateral and a triangle.

It should be noted that a polygon is a two-dimensional figure made up of a finite number of non-collinear consecutive segments, forming a closed space.

## Pentagon elements

Guiding us from the image below, the elements of the pentagon are the following:

- Vertices: A, B, C, D, E.
- Sides: AB, BC, CD, DE, AE.
- Interior angles: α, β, δ, γ, ε. They add up to 540º.
- Diagonals: Each interior angle is divided into three and there are five: AC, AD, BD, BE, CE.

## Pentagon types

We have two types of pentagon, according to their regularity:

- Regular: All its sides measure the same and also all its internal angles are equal and measure 108º, adding 540º. The two diagonals emerging from each vertex divide the corresponding internal angle into three equal parts measuring 36º (108º / 3).
- Irregular: Its sides have different lengths.

## Perimeter and area of a pentagon

To better understand the characteristics of a pentagon, we can calculate its perimeter and area:

- Perimeter (P): We add the sides of the polygon, that is: P = AB + BC + CD + DE + AE. If the pentagon is regular and all the sides have length L, it is true that P = 5L
- Area (A): We can also distinguish two cases. When it is an irregular pentagon, we could divide the figure into triangles, as we see in the image below. Thus, if we know the length of the diagonals, we can calculate the area of each triangle (as we explained in the triangle article) and do the summation.

In the example above, we could calculate the area of the triangles FGJ, GJI, and GHI.

Meanwhile, if the pentagon is regular, we can calculate the area based on the length of its side, following the following formula:

Likewise, we can calculate the area as a function of the apothem (which in the figure below is the QR segment), which is the segment that joins the center of a regular polygon with the midpoint of any of its sides, forming a right angle (which measures 90º). So the formula would be (where *to* the apothem and P the perimeter):

## Pentagon example

Suppose we have a regular pentagon with one side measuring 13 meters. What is the area and perimeter of the figure?

The perimeter would be:

P = 5 x 13 = 65 meters

Meanwhile, the area would be calculated as follows: