# Trapezoid

The trapezoid is a type of quadrilateral that does not have parallel sides. That is, as they are prolonged, the segments that make up the figure could intersect.

Unlike other quadrilaterals, the trapezoid does not have parallel sides. In addition, they can be distinguished from two types, the symmetric (or deltoid) and the asymmetric.

The symmetric trapezoid is one where two of the continuous sides measure the same, so it is said to be symmetric with respect to its diagonal. Thus, the crossing of the diagonals forms four right angles (90º).

In the lower image the symmetric trapezoid EF = FG and EH = GH

## Trapezoid elements

The elements of the trapezoid, as we can see in the following graphic, are the following:

- Vertices: A, B, C, D.
- Sides: AB, BC, DC, AD.
- Diagonals: AC, DB.
- Interior angles: α, β, δ, γ.

## Perimeter and area of a trapezoid

To better understand the trapezoid characteristics, we can calculate the perimeter and area:

- Perimeter (P): We must add the four sides of the quadrilateral.
- Area (A): Here we can distinguish two cases. First, when the trapezoid is asymmetric, we can divide the figure into two triangles (in the image below they would be triangle ABC and triangle ADC), calculate the area of each (as we explained in the triangle article) and add both data.

In the case of a symmetric trapezoid we will follow any of the following formulas where D and d are the lengths of the major and minor diagonal respectively. What's more, *to* and *b* are the lengths of the sides (remember that we have two pairs of sides that measure the same). Furthermore, α is the angle formed between two sides of different lengths.

## Trapezoid example

Suppose we have a symmetrical trapezoid where its sides measure 7 and 10 meters. Also, the angle formed between two sides that measure differently is 45º. What is the perimeter and area of the figure? (Take into account that being symmetric the trapezoid has two pairs of sides of equal length).

P = 7 + 7 + 10 + 10 = 24 m

Likewise, to calculate the area we use the second proposed formula:

A = 7 x 10 x sin (45º) = 49.4975 m2

## Other trapezoids

In the article, we have only mentioned the case of convex trapezoids, but we must mention that there are concave trapezoids, when any of the diagonals is external, as we see in the following image:

Likewise, we have that case of the crossed trapezoid when two of its sides intersect, forming two triangles, as we can see in the following graph:

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