The area of trapezoid : what is a trapezoid and what are its properties
A trapezoid is a quadrilateral figure whose both pairs of opposite sides are parallel to each other. Let us overlook the essential properties of a trapezoid as well as a formula for calculating the area of trapezoid :
 1. The middle line of a trapezoid is parallel to the bases and equal to the halfsum of them;
 2. The generalized theorem of Thales sounds as follows: the parallel lines intersecting sides of an angle of a trapezoid cut off proportional segments from the sides of the angle;
 3. Before we can proceed to the area of trapezoid we should know that in any equilateral trapezoid angles at the base are always equal;
 4. If we have an isosceles trapezoid then it is possible to describe a circle around it;
 5. If the sum of the bases of a trapezoid is equal to the sum of the sides, then it is possible to inscribe this figure in a circle (no matter what the area of trapezoid is);
 6. In a trapezoid the middles of the bases as well as the point of intersection of the diagonals and the continuation of the sides are always on the same straight line;
 8. The area of trapezoid can be found if to multiply 0.5 at the height of the trapezoid multiplied at the sum of lengths of the bases.
The area of trapezoid : concepts and ideas that underlie a genuine understanding of mathematics
How to become a mathematician? There are loads of opportunities to become interested in mathematics, probably, more than any other science can boast of, and many outstanding mathematicians showed unmistakable signs of a mathematical genius at a very early age. It is generally known that all children learn arithmetic and methods of calculating the area of trapezoid in elementary school – one does not need a special paper writing service for such tasks. This, of course, is a simple form of mathematics, but in such places great opportunities are hidden. For example, addition of integers can be a hint that there exists the problem of finding the sum of the series 1 + 2 + 3 +…. As a child, Gauss, who was one of the greatest mathematicians of the world, discovered a method of calculating the sum, which, indeed, is currently being used for summing arithmetic progressions. Moreover, all such wellknown concepts as line, triangle, square and the area of trapezoid are fraught with possibilities of geometric discoveries. There are rumors that Pascal discovered a part of Euclidean geometry independently at the age of twelve. Mathematics, namely algebra, elementary geometry and trigonometry that are usually taught in secondary schools in the United States and Canada, sometimes stimulate and invigorate the most talented pupils to solve tasks related, for example, to the area of trapezoid. However, the predominant part of the interest in mathematics are shown by students of the two said countries not earlier than on the second or third year of college education. In British middle schools, the teaching of analytic geometry and mathematical analysis has long been put in practice with an eye to leap forward students’ ability to think. In addition, these students usually get a good mathematical practice when preparing for entrance exams. By the time they go to the university, their attraction to mathematics and the desire to work in this field are quite outstandingly pronounced. It could be interesting to remind that Isaac Newton at the age of nineteen almost did not know mathematics (except, perhaps, for arithmetic) on the moment of his arrival at 1661 in Trinity College.
Sometimes, some tasks such as finding the area of trapezoid or very old mathematics textbooks having caught accidently the student’s eye, can instill her the desire to study the subject in more detail and with an exceptional eagerness. The imagination of the Indian mathematician Ramanujan was awakened by reading the most difficult chapters of his trigonometry textbook. It is almost inexplicable why such a book may attract a child, however, similarly to essay editing it makes an amazing impression on a child’s worldview as well as has an impact on the further development of a child’s brain. Her enthusiasm increases as she manages to understand the content of the book, in spite of all the difficulties and hindrances, even a little bit of success inspires her to apply further attempts. Quite often she manages to find quickly the area of trapezoid or solve the problem that requires a lot of time for her school friends or does not give in to them at all. She begins to understand the beauty of mathematics, more and more, and her interest is growing by leaps and bounds, and soon a huge variety of mathematics open to her. Then she cannot resist the attraction of this science and does math as her primary and most beloved occupation. Her decision becomes only stronger after passing various exams. However, some of the great mathematicians, such as Galois and Hermit, were not good, even terrible at exams. More recently, the evidence of the speculation on the approximation of algebraic numbers, which had been sought by many of the world strongest mathematicians, was provided by a man whose Cambridge gradebook was not as beautiful as one would expect for his outstanding results. Apparently, for the successful completion of examinations or calculating the area of trapezoid one needs some quality that may or may not go along with an inborn mathematical talent. On the other hand, an excellent exam grade is not always enough for detection of a brilliant future mathematician. It often happens that the one who gave great hopes to the parents in school and in college, demonstrates, in most cases, a helpless and dispirited behavior when serious creative problems stand up in front of them. For many people, such a situation is a key to the realization that mathematics is not an element they can be in. Thus, a number of students of the University of Cambridge, who were awarded honors diplomas for their thesis topics and the utmost importance of their works, have not done anything worthwhile in mathematics so far. But you will not go far in mathematics and not become a real mathematician if you do not have some of the necessary qualities. Thus, you necessarily need to have faith, hope and curiosity. You should constantly ask yourself: why, how and when, and this being of inquisitive bent must be the mainspring that moves you in your life. You must believe in your abilities, in your strength and always hope for success. Ultimately, you should never be discouraged even if you still cannot calculate the area of trapezoid, but always go forward and not allow yourself to become despondent no matter whatever hard task you are faced with.
The area of trapezoid : the historical peripeteia and stages of discovery
We want to emphasize once and for all that the term “mathematics” is used in this article exclusively in the sense of pure mathematics as they usually put this term in the UK; hence, pure math alongside with the term mathematician are set off against applied mathematics simultaneously with natural philosophy and theoretical physics. Our main interest in this largely unexplored science is associated with the results, theorems, methods and evidence of some of the theses, and not with practical applications of their consideration (such as whether they are suitable for calculating the area of trapezoid) or, alternatively, phenomena of the external world. Sometimes it is more useful, convenient and intuitive to express a mathematical problem in terms of material objects. But even when the material processes are studied, many people still remain predominantly interested in the mathematical side of things and nonabstract tasks, such as finding the area of trapezoid or knowing how to start a thesis ; such people should be assigned to the class of mathematicians. It is well known that interest and interest again was of a big asset to the fundamental development of pure mathematics. Others scientists and young researchers are much more worried about problems of the material world; they see the whole field of mathematics as a very valuable and powerful research tool. Obviously, one could say that one of the major problems of mathematics is to bring help to other sciences. It has become a common assertion that those sciences which fundamental results can be formulated mathematically and precisely can be developed and improved in the fastest way. Using mathematical methods we are able to display the most important consequences that hardly can be obtained in any other way. This thing alone, not to mention many other complicated aspects, such as the area of trapezoid, justifies the title of mathematics as the Queen of Sciences.
Even before their very early steps in building science humanity have learned to think about numbers conceiving them out of the context of those subjects of which they descend from. We will not make an attempt to define the area of trapezoid or the natural numbers 1, 2, 3, etc., instead, just repeat the phrase by Kronecker who said in his term paper : “God created the natural numbers. Everything else is the work of man.” Profoundly and indispensably, researchers require curiosity and an aptitude for observation and experimentation, needs to be able to cope with the power of reason and logic. Some properties of the natural numbers have been studied starting from a very early stage in human history. Euclid in his Book IX examines the properties of even and odd numbers, discovering, for example, that the sum of two even and two odd numbers is an even number too. But even purely practical problems (for instance, calculating the area of trapezoid ) were a great stimulus for studying further the properties of numbers. If you take a look at the wellknown result of Pythagoras’s calculations, which tells us about the connection between the length of sides of a right triangle. In the simplest case, both sides of a right triangle are 3:4:5, and it was an inexplicably useful result for the construction of a right angle. Inevitably, the ancient Greeks were faced with a question – do there exist any other right triangles with integer sides? Quite soon, they discovered a triangle with the aspect ratio of 5:12:13. Then we can see how it was going with the problem of finding the integers x, y, z, under such a condition that the equation x2 + y2 = z2 must be true. By using elementary mathematical transformations it can be shown that the solution is hidden in the following expressions: x = a2 – b2, y = 2ab, z = a2 + b2, where a and b are any integers. You can draw as many solutions of these expressions as you want. But now, of course, the next question is whether all the integral solutions that satisfy the aforementioned condition are covered by our formulas. Similarly to finding the area of trapezoid, this is a real mathematical problem, the solution of which is connected with the arithmetic properties of numbers, and even though the ancient Greeks were knowledgeable of these properties they did not solve the problem. The honor of finding the final solution belongs, apparently, to the unknown Arab mathematician who wrote the manuscript dating from the year 972 AD. There were other issues relating to squares and cubes, and many of the answers could be given by simply using the basic techniques that were pretty well known in those days. Many results were obtained in the fourth century AD by Diophantus who wrote a detailcrammed book on this topic. Under the Diophantine equation we now understand the equation f(x, y) = 0, especially if we are interested in its rational solutions, either whole or fractional. Just like with finding the area of trapezoid, the problem with this equation has a long history, and only in this century the most significant results related to it have been finally obtained.
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