He **decagon** It is a flat figure in the shape of a polygon with 10 sides and 10 vertices or points. Decagons can be regular or irregular, in the first case all the sides and internal angles have the same measure, while in the second the sides and/or angles are different from each other.

Examples of decagons of each type are shown in figure 1 and as we can see, the regular decagon is very symmetrical.

The basic elements of any decagon are:

-Sides, the line segments that when joined form the decagon.

-Vertices, or points between each consecutive side.

-Internal and external angles between adjacent sides.

-Diagonals, segments that join two non-consecutive vertices.

Vertices are named by capital letters, as shown in Figure 1, where the first letters of the alphabet were used, but any letter can be used.

The sides are symbolized with the two letters of the vertices between which they are found, for example the side AB is the one between the vertices A and B. The same is done with the diagonals, thus we have the diagonal AF, which joins points A and F.

For angles we use this symbol: ∠, similar to an inclined L. For example, the angle ∠ ABC is the one whose vertex is B and whose sides are the segments AB and BC.

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**regular decagon**

In the regular decagon, all the sides have the same measure, as well as the internal angles. Therefore it is said to be *equilateral* (equal sides) and *equiangular* (equal angles). It is a very symmetrical figure

**Interior angles of a regular decagon**

To find the measure of the internal angles of a regular polygon, including the regular decagon, the following formula is used:

Where:

-I is the measure of the angle in degrees.

-n is the number of sides of the polygon. In the case of the decagon n= 10.

Substituting n= 10 in the previous formula we obtain the following:

Now, a polygon is said to be *convex* if its angular measures are less than 180º, otherwise the polygon is *concave*. Since any internal angle of the regular decagon measures 144º and is less than 180º, then it is a convex polygon.

**sum of interior angles**

The sum of the measures of the interior angles of any polygon is, in degrees:

S = (n-2) x 180º; n is always greater than 2

In this formula we have:

-S is the sum of the measures of the interior angles.

-n is the number of sides. For the decagon n = 10

Applying the formula for n= 10 results:

S = (10 – 2) x 180º = 1440º

**exterior angles **

An exterior angle is formed between a side and the extension of the adjacent side, let’s see:

The angle ∠ ABC plus the exterior angle add up to 180º, that is, they are *supplementary*. Therefore the external angle is equal to 180º-144º = 36º, as we see in the figure.

**number of diagonals**

As said before, the diagonals are the segments that join non-consecutive vertices. How many diagonals can we draw in a decagon? When the number of vertices is small, they can be easily counted, but when that number increases, counting can be lost.

Fortunately, there is a formula to know the number of diagonals that a polygon of *no* sides:

For the decagon we substitute n = 10 and obtain:

D = 10 x (10 – 3) /2 = 35

In the regular decagon, all the diagonals intersect at a point, which is the center of the figure:

**Center**

The center of a polygon is defined as that point equidistant from any vertex. In the previous figure, the center coincides with the point of intersection of all the diagonals.

**Perimeter**

If the regular decagon has side a, its perimeter P is the sum of all the sides:

P = 10th

**Area**

Knowing the length *to* of the side, the area of the regular decagon is found by:

An approximate formula for area is:

And a third option to find the area is through the length of the apothem LA. This is the segment that joins the midpoint of one side to the center of the polygon.

In such a case the area can be calculated using the formula:

**irregular decagon**

The irregular decagon is not equilateral or equiangular, and generally lacks the symmetry of the regular figure, although some decagons may have a line of symmetry.

They can also be convex or concave, if there are internal angles greater than 180º.

The irregular decagon in figure 1 is concave, since some of its internal angles are greater than 180º. It is evident that there are many combinations of angles and sides that give rise to an irregular decagon.

In any case, it is true that:

-The internal angles of an irregular decagon also add up to 1440º.

-It also has 35 diagonals.

**Area of an irregular decagon by Gaussian determinants**

In general, there is no single formula for finding the area of an irregular polygon, since the sides and angles are different. However, it can be found by knowing the coordinates of the vertices and calculating the *Gaussian determinants*:

-Let’s call (xn , yn ) the coordinates of the vertices, with *no* ranging from 1 to 10.

-You can start from any vertex, to which the coordinates (x1, y1 ) will be assigned. Now we have to substitute the values of each coordinate in this formula:

Where the determinants are precisely the operations between parentheses.

-It is important to note that the last determinant again involves the first vertex together with the last one. For the decagon, it would look like this:

(x10y1 – x1y10)

**Important:** The bars are those of absolute value and mean that the final result is always given with a positive sign.

The procedure can be laborious when the figure has many vertices, in the case of the decagon there are 10 operations, so it is advisable to make a table or a list.

**solved exercise**

Calculate the area of the irregular decagon shown in the figure. The coordinates of the vertices are A, B, C… J, whose values are shown on the left.

**Solution**

-We do each of the 10 operations:

2×6 – 4×0 = 12 – 0 =12 0×4 – 6×(-2) = 0 + 12 =12 (-2)×7- 4×(-5) = -14 + 20 = 6 ( -5)×2 – 7×(-6) = -10 + 42 = 32 (-6)×(-4) – 2×(-4) = 24 + 8 =32 (-4)×(-2) – (-4)×(-2) = 8 – 8 =0 (-2)×0 – (-2)×(-1) =0 -2 (-1)×0 – 0×(2) = 0 – 0 = 0 2×2 – 0×8 = 4 – 0 = 4 8×4 -2×2 = 32 – 4 = 28

-We add the results:

12 + 12 + 6 + 32 + 32 + 0 + (-2) + 0 + 4 + 28 = 124

A positive result is obtained even without the absolute value bars, but if it is negative, the sign is changed.

-The previous result is divided by 2 and that is the area of the polygon:

A=124/2 = 62

**decagon properties**

Below is a summary of the general properties of a decagon, whether regular or irregular:

-It has 10 sides and 10 vertices.

-The sum of the internal angles is 1440º.

-There are 35 diagonals.

-The perimeter is the sum of all the sides.

-Triangles can be created inside a polygon by drawing segments from one vertex to all the others. In a decagon it is possible to draw 8 triangles in this way, as in the one shown below:

**References**

Alexander, D. 2013. Geometry. 5th. Edition. Cengage Learning. decagon.com. Decagon. Retrieved from: decagono.com Math Open Reference. decagon. Retrieved from: mathopenref.com. Sangaku Maths. Elements of a polygon and their classification. Recovered from: sangakoo.com. Wikipedia. Decagon. Recovered from: es.wikipedia.com.