RIGHT TRAPEZOID: PROPERTIES, RELATIONSHIPS AND FORMULAS, EXAMPLES - MATHS - 2024

A right 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" that is given to the figure. The following image of a right trapezoid clarifies these characteristics:

Trapezoid elements

The elements of the trapezoid are:

-Bases

-Vertices

-Height

-Internal angles

-Middle base

-Diagonals

We are going to detail these elements with the help of figures 1 and 2:

Figure 1. A right trapezoid, characterized by having two 90º internal angles: A and B. Source: F. Zapata.

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

According to the definition, the bases of this trapezoid are the sides a and b, which as observed are parallel and also have different lengths.

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

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

The median base is the segment joining the midpoints of the non-parallel sides (segment EF in Figure 2).

Figure 2. The elements of the right trapezoid. Source: F. Zapata.

And finally there are the diagonals d ₁ and d ₂, the segments that join the opposite vertices and that intersect at point O (see figure 2).

Relationships and formulas

Trapezoid height h

Perimeter P

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

Side d is expressed in terms of the height or side c by the Pythagorean theorem:

Substituting in the perimeter:

Middle base

It is the semi-sum of the bases:

Sometimes the mean base is found expressed like this:

Area

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

Diagonals, sides and angles

In Figure 2 several triangles appear, both right and non-right. 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 d ₁, therefore:

DAB triangle

It is also a rectangle, the legs are a and c (or also ayh) and the hypotenuse is d ₂, so that:

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:

CDP triangle

This triangle is a right triangle and with its sides the trigonometric ratios of the angle α are constructed:

But the side PD = a - b, therefore:

You also have:

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 highlighted that it is 180 - α. This triangle is not a right triangle, so the cosine theorem or the sine theorem can be applied.

Now, it can easily be shown that:

Applying the cosine theorem:

Examples of right trapezoids

Trapezoids and in particular right trapezoids are found on many sides, 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 many 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), plates and in graphic design.

Figure 3. Angel inside a rectangle trapezoid in a New York church. Source: David Goehring via Flickr.

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.

Figure 4. A trapezoidal signal. Source: Wikimedia Commons.

In numerical calculation

To compute in numerical form the definite integral of the function f (x) between a and b, the trapezoid rule is used 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 the one in the right figure, with multiple right trapezoids.

Figure 5. A definite integral between a and b is nothing other than the area under the curve f (x) between these values. A right trapezoid can serve as a first approximation for such an area, but the more trapezoids used, the better the approximation. Source: Wikimedia Commons.

Beam with trapezoidal load

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, surface 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:

Figure 6. Loads on a beam. Source: Bedford, A. 1996. Static. Addison Wesley Interamericana.

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

As an educational and learning tool

Geometric-shaped blocks and pictures, including trapezoids, are very helpful in acquainting children with the fascinating world of geometry from an early age.

Figure 7. Blocks with simple geometric shapes. How many right trapezoids are hidden in the blocks? Source: Wikimedia Commons.

Solved exercises

- Exercise 1

In the right trapezoid in 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 is 35 cm. Find:

a) Angle α

b) Height

c) Perimeter

d) Average base

e) Area

f) Diagonals

Solution to

The statement data is summarized as follows:

a = larger base = 50 cm

b = smaller base = 30 cm

d = slant side = 35 cm

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

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

Thus:

It clears h:

d ₁ ² = 2 x (30 cm) ² = 1800 cm ²

d ₁ = √1800 cm ² = 42.42 cm

And for the diagonal d ₂:

References

1. Baldor, A. 2004. Plane and space geometry with trigonometry. Cultural Publications.
2. Bedford, A. 1996. Statics. Addison Wesley Interamericana.
3. Jr. geometry. 2014. Polygons. Lulu Press, Inc.
4. OnlineMSchool. Rectangular trapezoid. Recovered from: es.onlinemschool.com.
5. Automatic geometry problem solver. The trapeze. Recovered from: scuolaelettrica.it
6. Wikipedia. Trapezoid (geometry). Recovered from: es.wikipedia.org.