A quadrangle with right angles is ... The sum of the angles of a quadrilateral

One of the most interesting topics on geometry from the school course is "Quadrangles" (8th grade). What kinds of such figures exist, what special properties do they have? What is the uniqueness of quadrangles with angles of ninety degrees? Let's look into all this.

What geometric figure is called a quadrangle

Polygons, which consist of four sides and, respectively, of four vertices (angles), are called quadrilaterals in Euclidean geometry.

The history of the name of this kind of figures is interesting. In Russian, the noun "quadrangle" is formed from the phrase "four corners" (in the same way as "triangle" - three corners, "pentagon" - five corners, etc.).

However, in Latin (through whose mediation many geometric terms have come in most of the world's languages) it is called quadrilateral. This word is formed from the numeral quadri (four) and the noun latus (side). So we can conclude that in ancient this polygon was called just "quadrilateral".

By the way, this name (with the emphasis on the presence of four sides, not angles) in this figure has been preserved in some modern languages. For example, in English - quadrilateral and in French - quadrilatère.

At the same time, in most Slavic languages the type of figures under consideration is still identified by the number of angles, not by the sides. For example, in Slovak (štvoruholník), in Bulgarian ("chetryriaglnik"), in Belarusian ("chatyrokhkutnik"), in Ukrainian ("chotirikutnik"), in Czech (čtyřúhelník), but in Polish the quadrangle is named by the number of parties - czworoboczny.

What kinds of quadrangles are studied in the school curriculum

In modern geometry four types of polygons with four sides are distinguished. However, because of the too complicated properties of some of them, only two species are introduced to schoolchildren in geometry classes.

• Parallelogram (parallelogram). The opposite sides of a quadrilateral are pairwise parallel to each other and, respectively, are also pairwise.
• Trapezium (trapezium or trapezoid). This quadrilateral consists of two opposite sides, parallel to each other. However, the other pair of parties does not have such a feature.

Types of quadrangles not studied in the school course of geometry

In addition to the above, there are two more types of quadrangles that schoolchildren are not introduced to in geometry lessons, because of their particular complexity.

• The deltooid (kite) is a figure in which each of the two pairs of adjacent sides is equal in length to each other. The name of this quadrangle is due to the fact that in appearance it is quite similar to the letter of the Greek alphabet - "delta".
• Antiparallelogram - this figure is just as complex as its name. In it, the two opposite sides are equal, but they are not parallel to each other. In addition, the long opposite sides of this quadrilateral intersect, like the continuations of the other two shorter sides.

Types of parallelogram

Having dealt with the main types of quadrangles, it is worth paying attention to its subspecies. Thus, all parallelograms, in turn, are also divided into four groups.

• Classical parallelogram.
• Rhombus (rhombus) is a quadrangular figure with equal sides. Its diagonals intersect at a right angle, dividing the diamond into four equal rectangular triangles.
• Rectangle. The name speaks for itself. Since it is a quadrilateral with right angles (each of them is equal to ninety degrees). Opposite sides of it are not only parallel to each other, but also equal.
• Square (square). Like a rectangle, it is a quadrilateral with right angles, but all sides are equal to each other. This figure is close to the rhombus. So it can be argued that a square is a cross between a diamond and a rectangle.

Special properties of the rectangle

Considering the figures in which each of the angles between the sides is equal to ninety degrees, it is worth to look more closely at the rectangle. So, what special features does he have that distinguish him from other parallelograms?

To assert that the parallelogram in question is a rectangle, its diagonals must be equal to each other, and each of the angles is straight. In addition, the square of its diagonals must correspond to the sum of the squares of the two adjacent sides of this figure. In other words, the classical rectangle consists of two rectangular triangles, and in them, as is known, the sum of the squares of the legs is equal to the square of the hypotenuse. In the role of the hypotenuse, the diagonal of the quadrilateral in question appears.

The last of the listed characteristics of this figure is also its special property. In addition, there are others. For example, the fact that all sides of the quadrilateral with straight angles is at the same time its heights.

In addition, if you draw a circle around any rectangle, its diameter will be equal to the diagonal of the inscribed figure.

Among other properties of this quadrilateral, the fact that it is flat and in non-Euclidean geometry does not exist. This is due to the fact that in this system there are no quadrangular figures whose sum of angles is three hundred and sixty degrees.

Square and its features

Having dealt with the characteristics and properties of the rectangle, it is worth paying attention to the second known quadrangle with straight angles (this is a square).

Being in fact the same rectangle, but with equal sides, this figure has all its properties. But unlike it, the square is present in non-Euclidean geometry.

In addition, this figure, there are other own distinctive features. For example, the fact that the diagonals of a square are not simply equal to one another, but also intersect at a right angle. Thus, like a diamond, the square consists of four rectangular triangles, into which it is divided by diagonals.

In addition, this figure is the most symmetrical among all the quadrangles.

What is the sum of the angles of a quadrilateral

Considering the singularities of quadrilaterals of Euclidean geometry, it is worth paying attention to their angles.

So, in each of the above figures, regardless of whether it has right angles or not, their total sum is always the same - three hundred and sixty degrees. This is a unique feature of this kind of figure.

Perimeter of quadrilaterals

Having dealt with what is the sum of the angles of a quadrilateral and other special properties of figures of this kind, it is worthwhile to find out which formulas are best used to calculate their perimeter and area.

To determine the perimeter of any quadrilateral, you only need to add the length of all its sides.

For example, in the figure KLMN its perimeter can be calculated by the formula: P = KL + LM + MN + KN. If we substitute the numbers here, we get: 6 + 8 + 6 + 8 = 28 (cm).

In the case where the figure in question is a rhombus or a square, to find the perimeter, you can simplify the formula by simply multiplying the length of one of its sides by four: P = KL x 4. For example: 6 x 4 = 24 (cm).

Square quadrangle formulas

Having dealt with how to find the perimeter of any figure with four corners and sides, it is worthwhile to consider the most popular and simple ways of finding its area.

• The classical way of calculating it is to use the formula S = 1/2 KM x LN x SIN LON. It turns out that the area of any quadrilateral is equal to half the product of its diagonals by the sine of the angle located between them.
• If the figure whose area you want to find is a rectangle or a square (diagonals are always equal to each other), you can simplify the formula by squaring the length of one diagonal and multiplying it by the sine of the angle between them and dividing everything in half. For example: S = 1/2 KM ² x SIN LON.
• Also, when finding the area of a rectangle, information on the perimeter of the figure in question and the length of one of its sides can help. In this case, it will be most convenient to use the formula S = KN x (P - 2 KN) / 2.
• In the case of a square, its properties allow us to use several additional formulas to find the area. For example, knowing the perimeter of a figure, you can use this option: S = P ² / 16. And if the radius of a circle inscribed in a quadrilateral is known, the square of the square is in a very similar way: S = 4r ² . If the radius of the circumscribed circle is known, then another formula is suitable: S = 2R ² . Also, the area of the square is 0.8 times the length of the line drawn from the corner of the figure to the middle of the opposite side.
• In addition to all the above, there is also a separate formula for finding the area, calculated specifically for the parallelogram. It can be used, if known, the length of two heights of the figure and the size of the angle between them. Then the heights must be multiplied between themselves and the sine of the angle between them. It is worth noting that you can use this formula for all figures that relate to parallelograms (that is, to a rectangle, a rhombus and a square).

Other properties of quadrangles are inscribed and circumscribed circles

Having considered the features and properties of a quadrangle as a figure of Euclidean geometry, it is worthwhile to pay attention to the possibility of describing around or inscribing circles inside it:

• If the sums of the opposing corners of the figure are one hundred and eighty degrees and are pairwise equal, then around this quadrilateral one can freely describe a circle.
• According to Ptolemy's theorem, if a circle is described outside the polygon with four sides, then the product of its diagonals is equal to the sum of the products of opposite sides of this figure. Thus, the formula will look like this: KM x LN = KL x MN + LM x KN.
• If we construct a quadrilateral in which the sums of the opposite sides are equal to each other, then a circle can be inscribed into it.

Having dealt with what a quadrilateral is, what kinds of it exist, which of them have only direct angles between the sides and what properties they possess, it is worth remembering all this material. In particular, the formulas for finding the perimeter and area of the polygons examined. After all, figures of this form - one of the most common, and this knowledge can be useful for calculations in real life.