# Quadrilateral

The quadrilateral is a geometric figure, specifically a polygon, made up of four sides, four angles, and four vertices.

It should be noted that a polygon is a closed two-dimensional figure made up of a finite number of consecutive segments. The segments are called sides and their intersections, vertices.

The quadrilateral is then a figure with four sides, which may or may not be of equal length. It also has four interior and exterior angles, corresponding to each vertex.

In addition, each quadrilateral has two diagonals, which are those segments that join one side or vertex of a geometric figure with the opposite side.

## Quadrilateral elements

Guiding us from the graph at the bottom, the quadrilateral elements are as follows:

- Vertices: A, B, C, D.
- Sides: AB, BC, DC, AD.
- Interior angles: w, x, y, z. They add up to 360º.
- Exterior angles: s, t, u, v.
- Diagonals: They are the line segments that join opposite vertices of the figure. They are AC and DB.

## Quadrilateral types

The types of quadrilateral are:

- Parallelogram: It is a quadrilateral where the opposite sides are parallel to each other (the segments would not intersect even if they were prolonged) and measure the same length. It is a category within which there are several others.
- Square: It is a type of parallelogram with four sides of equal length and parallel to each other. Its interior angles are right, that is, they measure 90º. Their diagonals are perpendicular to each other (when they intersect they form four 90º angles).
- Rectangle: Of its four sides, there are two pairs of sides of equal length. All its interior angles measure 90º. Their diagonals measure the same, but they are not perpendicular to each other.
- Rhombus: All its sides are the same length. Two of its interior angles are acute (less than 90º), they measure the same and are opposite each other. Meanwhile, the other two interior angles are obtuse (greater than 90º) and also measure the same. Their diagonals are perpendicular to each other, but they measure differently.
- Rhomboid: It has two pairs of sides that correspond in length and has two acute and two obtuse interior angles. Each pair of angles, which also measure the same, are facing each other.

- Trapezoid: It has only two sides that are parallel to each other, called the base of the trapezoid, and that have different lengths. The height of the trapezoid is the line segment that joins both bases or their extensions.
- Trapezoid: It is a quadrilateral without parallel sides.

Quadrilaterals can also be classified based on the measure of their angles:

- Concaves: When at least one of its interior angles is greater than 180 °.
- Convex: When none of its interior angles measures more than 180 °.

## Perimeter and area of the quadrilateral

To better understand the characteristics of a quadrilateral, we can calculate the following:

- Perimeter (P): It is the sum of the sides:

P = AB + BC + CD + AD

- Area (A): The calculation complexity varies in each case. In a square, for example, only the length of the side is squared. However, a formula that applies to all types of quadrilateral can be applied:

Where s is the semiperimeter (P / 2), and α y β are two opposite angles of the quadrilateral. Also, a, b, c, and d are the lengths of the sides, and cos indicates that the cosine of an angle will be calculated.

## Quadrilateral example

Suppose we have a quadrilateral whose sides and their respective lengths are as follows (all measured in meters):

AB: 23

BC: 10

AC: 25

AD: 12

Likewise, the angle formed between AB and BC is 40º and that between CD and AD is 60º. What is the perimeter and area of the quadrilateral?

P = 23 + 10 + 25 + 12 = 70 meters

So, to calculate the area, we first find the semiperimeter and apply the formula shown in the previous section:

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