A **rectangle trapezoid** is a flat figure with four sides, such that two of them are parallel to each other, called *bases* and also one of the other sides is perpendicular to the bases.

For this reason, two of the internal angles are right, that is, they measure 90º. Hence the name «rectangle» given to the figure. The following image of a rectangular trapezoid clarifies these characteristics:

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**elements of the trapezoid**

The elements of the trapezoid are:

-Bases

-Vertices

-Height

-internal angles

-Medium foundation

-Diagonals

We will detail these elements with the help of figures 1 and 2:

The sides of the right trapezoid are denoted by the lowercase letters a, b, c, and d. The corners of the figure or *vertices* are indicated in capital letters. Finally the *internal angles* They are expressed with Greek letters.

By definition, the *bases* of this trapezoid are the sides a and b, which as can be seen are parallel and also have different lengths.

The side perpendicular to both bases is the side *c* to the left, which is *height* *h* of the trapezoid And finally there is the side d, which forms the acute angle α with the side a.

The sum of the *internal angles* of a quadrilateral is 360º. It is easy to see that the missing angle C in the figure is 180 – α.

The *medium foundation* is the segment that joins the midpoints of the non-parallel sides (segment EF in figure 2).

And finally there are the diagonals d1 and d2, the segments that join the opposite vertices and that intersect at point O (see figure 2).

**relations and formulas**

**height h of the trapezoid**

*h=c*

**perimeter P**

It is the measurement of the contour and is calculated by adding the sides:

*Perimeter = a + b + c + d*

The side *d* is expressed in terms of height or side *c* Using the Pythagorean theorem:

*d = √(ab)2 + c2 *

Substituting in the perimeter:

*P = a + b + c + √(ab)2 + c2*

**medium foundation**

It is the half sum of the bases:

*Mean basis = (a+b)/2*

Sometimes the average base is expressed in this way:

*Middle Base = (Major Base + Minor Base) /2*

**Area**

The area A of the trapezoid is the product of the median base times the height:

*to =* *(Major base + minor base) x height /2*

*A = (a+b)c/2*

**Diagonals, sides and angles**

Several triangles appear in figure 2, both right and non-rectangle. The Pythagorean theorem can be applied to those that are right triangles and to those that are not, the cosine and sine theorems.

In this way relationships are found between the sides and between the sides and internal angles of the trapezoid.

**CPA triangle**

It is a rectangle, its legs are equal and are worth b, while the hypotenuse is the diagonal d1, therefore:

*d12 = b2 + b2 = 2b2*

**DAB triangle**

It is also a rectangle, the legs are *to* and *c* (or also *to* and *h*) and the hypotenuse is d2, so that:

*d22 = a2 + c2 = a2 + h2*

**CDA triangle**

Since this triangle is not a right triangle, the cosine theorem is applied to it, or also the sine theorem.

According to the cosine theorem:

*d12 = a2 + d2 – 2ad cos α*

**CDP triangle**

This triangle is right-angled and with its sides the trigonometric ratios of the angle α are built:

*sin α = h/d*

*cos α = PD/d*

But the side PD = a – b, therefore:

*cos α = (ab) / d → a – b = d cos α*

*a = b + d cos α*

You also have:

*tg α = sin α / cos α = h / (ab) → h = tg α (ab)*

**CBD triangle**

In this triangle we have the angle whose vertex is at C. It is not marked in the figure, but at the beginning it was noted that it is worth 180 – α. This triangle is not a right triangle, so the theorem of cosines or theorem of sines can be applied.

Now, it can be easily shown that:

*sin (180 – α) = sin α*

*cos (180 – α) = – cos α*

Applying the cosine theorem:

*d22 = d2 + b2 – 2db cos (180 – α) = d2 + b2 + 2db cos α*

**Examples of right trapezoids**

Trapezoids and in particular rectangular trapezoids are found in many places, and sometimes not always in tangible form. Here we have several examples:

**The trapezoid as a design element**

Geometric figures abound in the architecture of numerous buildings, such as this church in New York, which shows a structure in the shape of a rectangular trapezoid.

Likewise, the trapezoidal shape is frequent in the design of containers, containers, blades (*cutter* or exact), badges and in graphic design.

**trapezoidal wave generator**

Electrical signals can not only be square, sinusoidal or triangular. There are also trapezoidal signals that are useful in many circuits. In figure 4 there is a trapezoidal signal composed of two right trapezoids. Between them they form a single isosceles trapezoid.

**In numerical calculation**

To compute the definite integral of the function f(x) between a and b numerically, we use the trapezoidal rule to approximate the area under the graph of f(x). In the following figure, on the left, the integral is approximated with a single right trapezoid.

A better approximation is that of the right figure, with multiple right trapezoids.

**trapezoidal loaded beam**

Forces are not always concentrated on a single point, since the bodies on which they act have appreciable dimensions. Such is the case of a bridge over which vehicles circulate continuously, the water of a swimming pool on the vertical walls of the same or a roof on which water or snow accumulates.

For this reason, forces are distributed per unit of length, area or volume, depending on the body on which they act.

In the case of a beam, a force distributed per unit length can have various distributions, for example the right trapezoid shown below:

In reality, distributions do not always correspond to regular geometric shapes like this, but they can be a good approximation in many cases.

**As an educational and learning tool**

Blocks and sheets with geometric shapes, including trapezoids, are very useful for children to become familiar with the fascinating world of geometry from an early age.

**solved exercises**

**– Exercise 1**

In the rectangular trapezoid of figure 1, the larger base is 50 cm and the smaller base is equal to 30 cm, it is also known that the oblique side measures 35 cm. Find:

a) Angle α

b) Height

c) Perimeter

d) Average basis

e) Area

f) Diagonals

**Solution to**

The statement data is summarized as follows:

a = larger base = 50 cm

b = minor base = 30 cm

d = inclined side = 35 cm

To find the angle α we visit the formulas and equations section, to see which is the one that best adapts to the data offered. The angle sought is found in several of the triangles analysed, for example the CDP.

There we have this formula, which contains the unknown and also the data that we know:

*cos α = (ab) / d*

Therefore:

*α = arcs [(a-b) / d]* = arcs [(50-30) / 35 ] = arcs 20/35 = 55.15º

**solution b**

From the equation:

*sin α = h/d*

It clears h:

*h = d.sin α* = 35 sin 55.15 º cm = 28.72 cm

**solution c**

The perimeter is the sum of the sides, and since the height is equal to side c, we have:

c = h = 28.72 cm

Therefore:

P = (50 + 30 + 35 + 28.72) cm = 143.72 cm

**solution d**

The average base is the half sum of the bases:

Average base = (50 + 30 cm )/2 = 40 cm

**solution and**

The area of the trapezoid is:

A = middle base x height = 40 cm x 28.72 = 1148.8 cm2.

**Solution f**

For the diagonal d1 you can use this formula:

* **d12 = b2 + b2 = 2b2*

d12= 2 x (30 cm)2 = 1800 cm2

d1 = √1800 cm2 = 42.42 cm

And for the diagonal d2:

*d22 = d2 + b2 + 2db cos α* = (35 cm)2 + (30 cm)2 + 2 x 35 x 30 cm2 cos 55.15 º = 3325 cm2

d2 = √ 3325 cm2 = 57.66 cm

This is not the only way to find d2, as there is also triangle DAB.

**– Exercise 2**

The following graph of velocity as a function of time belongs to a mobile that has uniformly accelerated rectilinear motion. Calculate the distance traveled by the mobile during the time interval between 0.5 and 1.2 seconds.

**Solution**

The distance traveled by the mobile is numerically equivalent to the area under the graph, delimited by the indicated time interval.

The shaded area is the area of a right trapezoid, given by:

*to =* *(Major base + minor base) x height /2*

*A = (1.2 + 0.7) m/s x (1.2 – 0.5) s/2 = 0.665 m*

**References**

Baldor, A. 2004. Plane and space geometry with trigonometry. Culture Publications.

Bedford, A. 1996. Statics. Addison Wesley Interamericana.

Jr. geometry. 2014. Polygons. Lulu Press, Inc.

OnlineMSchool. Rectangle trapezoid. Retrieved from: es.onlinemschool.com. Automatic geometry problem solver. the trapezoid Retrieved from: scuolaelettrica.it Wikipedia. Trapezoid (geometry). Recovered from: es.wikipedia.org.