Diagonals of a polygon

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The diagonals of a polygon are those segments that join vertex with its opposite vertex (s).

The diagonals of a polygon are then those lines that start from one vertex and end at another, and there may be more than one diagonal per vertex.

For example, in the square below, the diagonals are segments AC and BD.

Square and its diagonals Diagonal of a square Diagonal of a rectangle

It is worth remembering that the vertex of a polygon is that point where two consecutive sides of the figure meet.

Likewise, a polygon is a two-dimensional figure made up of a finite series of continuous, non-collinear segments that form a closed space.

It is important to specify that the diagonals of a polygon may or may not be of the same length. For example, in the case of the rhombus, it has a major and a minor diagonal.

It is worth adding, in addition, that the only polygon that does not have diagonals is the triangle.

How to calculate the number of diagonals in a polygon

To calculate the number of diagonals (N) of a polygon, from the number of sides it has (n), we can use the following formula:

This equation can be interpreted as follows → Each vertex of the polygon has a number of diagonals which is the number of sides minus three or n-3 (remember that the number of vertices is equal to the number of sides). The diagonal does not join the vertex with itself or with the two contiguous vertices. Likewise, in order not to count the same diagonal twice, the division is made by two.

Exercises with the diagonals of the polygon

Let's look at some exercises. How many diagonals does a polygon with nine sides have? Applying the formula shown above, we would solve as follows:

That is, an eneagon has 27 diagonals.

Now, suppose we know that the polygon has 44 diagonals, and what we need to find is the number of sides:

We solve the quadratic equation and, since the number of sides cannot be negative, the answer is eleven.

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