# Eneagon

The eneagon or nonagon is a geometric figure with nine sides. It also has nine vertices and nine interior angles.

That is, the enegon is a polygon that has nine sides, so it is more complex than an octagon or a heptagon.

It should be remembered that a polygon is a two-dimensional (two-dimensional) figure made up of a set of consecutive segments that do not belong to the same line, and that form a closed space.

## Elements of the eneagon

Taking the image below as a reference, the elements of the enegon are the following:

- Vertices: A, B, C, D, E, F, G, H, I.
- Sides: AB, BC, CD, DE, EF, FG, GH, HI and AI.
- Interior angles: α, β, δ, γ, ε, ζ, η, θ, i. They add up to 1260º.
- Diagonals: There are 27 and they start at 5 of each interior angle: AC, AD, AE, AF, AG, AH, BD, BE, BF, BG, BH, BI, CF, CG, CE, CH, CI, DF, DG , DH, DI, EG, EH, EI, FH, FI, GI.

## Eneagon types

According to their regularity, we have two types of eneagons:

- Irregular: Their sides (and their internal angles) are not equal, at least one differs.
- Regular: Their sides measure the same, like their interior angles that are each one of 140º.

## Perimeter and area of the enegon

To better understand the characteristics of the enegon, we can follow the following formulas:

- Perimeter (P): We add the sides of the figure: P = AB + BC + CD + DE + EF + FG + GH + HI + AI. If the enegon is regular, just multiply the side length (L) by 9: P = 9xL
- Area (A): Let's look at two cases. First, when the figure is irregular, it can be divided into several triangles (see image below). If we know the length of the diagonals drawn, we can calculate the area of each triangle (following the steps we explained in the triangle article) and then do the summation.

In a second case, if the enegon is regular, we multiply the perimeter by the apothem (a) and divide it by two, as we see in the following formula:

The apothem is defined as the line that joins the center of a regular polygon with the midpoint of any of its sides. Between the apothem and the side of the polygon, a right angle is formed (measuring 90º). Then, it is possible to express the apothem as a function of the length of the side of the enegon.

First, let's observe in the image above that the central angle (α) in the eneagon is equal to the division of 360º by 9, that is, 40º. Next, we note that triangle SJT is a right triangle (S is the midpoint of the polygon). The hypotenuse is SJ, one leg is L / 2 (half the length of the side), and the other leg is apothem (a). Similarly, α / 2 is 20º (40/2). So, let's remember that the tangent (tan) of the angle of a right triangle is equal to the opposite leg (L / 2) between the adjacent leg that is apothem (a) and we solve it as follows, taking as reference the angle α /2:

Then we plug in a into the formula for the area. In this way, we will have the equation as a function of L (the side of the enegon):

## Eneagon example

Suppose we have a regular enegon with a length of its sides of 18 meters. What is the perimeter and area of the polygon?

Therefore, the area of this enegon is 2002.9110 m2 and the perimeter is 162 meters.