What is a square? How to find the vertices, section, plane, equation, volume, base area and angle of the square?

The answers to the question about what a square is, can be a lot. It all depends on who you addressed this question to. The musician will say that the square is 4, 8, 16, 32 bars or jazz improvisation. A child is a game with a ball or a children's magazine. The printer will send you to study the skittles of the font, and the technician - a variety of metal rolling profile.

There are many other meanings for this word, but today we will ask a question to mathematics. So...

We will gradually deal with this figure, from simple to complex, and begin with the history of the square. How did it come about, how did people perceive it, scientists of different countries and civilizations?

History of the study of the square

The ancient world perceives the square, mainly as the four sides of the world. In general, despite the many quadrangles, it is at the square that the main number is four. For the Assyrians and Peruvians, the square is the whole world, that is, it represents the four main directions, the directions of the world.

Even the universe was represented as a square, also divided into four parts - this is the vision of the inhabitants of North America. For the Celts, the universe is as many as three squares nested inside each other, and four (!) Rivers flow from the center. And the Egyptians generally deified this figure!

For the first time, the square was described using the mathematical formulas of the Greeks. But for them, this polygon had only negative characteristics. Pythagoras generally did not like even numbers, seeing in them weakness and femininity.

Even in religions there is a square. In Islam, the Kaaba - the navel of the Earth - has not some spherical, but a cubic form.

In India, the main grapheme depicting the Earth, or the symbol of the earth, was a crossed square. And again we are talking about the four sides of the world, the four regions of the earth.

In China, the square is peace, harmony and order. Chaos is defeated by building a square Varya. And the square inscribed in the circle is the basis of the vision of the world, symbolizing the unity and connection of the Cosmos and the Earth.

Pagan Russia - Svarog Square. This symbol is also called the Star of Svarog, or the Star of Russia. It is quite complex, because it is made up of intersecting and closed lines. Svarog - the god the Blacksmith, the most important creator, the creator and the sky itself in the representation of Rusich. In this symbol there is a rhombus, which again speaks about the Earth and its four directions. And a star with four rays - 4 sides of the world, 4 faces of Svarog - his omniscience. And the intersection of the rays is the hearth.

Interesting about the square

The most popular phrase that comes to mind about our main character is "Black Square".

Painting by Malevich is still very popular. The author himself after his creation was long tormented by the question of what it is and why a simple black square on a white background so draws attention to itself.

But if you look closely, you will notice that the plane of the square is not smooth, but in the cracks of the black paint there are many multicolored shades. Apparently, at first there was a certain composition that the author did not like, and he closed it from our eyes with this figure. Black square, like nothing - a black hole, only a magical square shape. And emptiness, as you know, attracts ...

Still very popular are the "magic squares". In fact, this is a table, of course, square, filled with numbers in each column. The sum of these numbers is the same in all rows, columns and diagonals (separately). If the diagonals are excluded from equality, then the square is semi-magical.

Albrecht Durer in 1514 created a picture of "Melancholy I", which depicted a magic square 4x4. In it, the sum of the numbers of all the columns, rows, diagonals and even inner squares is thirty-four.

On the basis of these tables appeared very interesting and popular puzzles - "Sudoku".

The Egyptians were the first to conduct lines of interrelation of numbers (date of birth) and qualities of character, abilities and talents of a person. Pythagoras took this knowledge, revised it several times and placed it squarely. It turned out the Pythagoras Square.

This is already a separate direction in numerology. From the date of birth of a person, by addition, four basic numbers are calculated, which are placed in the psychomatrix (square). So lay out all the secret information about your energy, health, talent, luck, temperament and other things on the shelves. On average, according to polls, the reliability is 60% -80%.

What is a square?

A square is a geometric figure. The shape of the square is a quadrilateral, which has equal sides and angles. More precisely, this quadrangle is called correct.

The square has its signs. It:

• Sides equal in length;
• Equal angles are straight (90 degrees).

By virtue of these characteristics and features, a circle can be inscribed in a square and described around it. The circumscribed circle will touch all its vertices, inscribed - the middle of all its sides. Their center will coincide with the center of the square and divide all its diagonals in half. The latter, in turn, are equal to each other and divide the corners of the square into equal parts.

One diagonal divides the square into two isosceles triangles, and two - to four.

Thus, if the length of the side of the square is t, the radius of the circumscribed circle is R, and the inscribed length is r, then

• The square of the base of the square, or the area of the square (S) will be S = t ² = 2R ² = 4r ² ;

• The perimeter of the square P should be calculated by the formula P = 4t = 4√2R = 8r;

• The length of the radius of the circumscribed circle R = (√2 / 2) t;

• Inscribed - r = t / 2.

The area of the base of the square can still be calculated, knowing its side (a) or the length of its diagonal (c), then the formulas will look like: S = a ² and S = 1 / 2c ² .

What is a square, we found out. Let's take a closer look at the details, because the square figure is the most symmetrical quadrilateral. It has five axes of symmetry, one (of the fourth order) passing through the center and being perpendicular to the plane of the square itself, and four others - symmetry axes of the second order, two of them are parallel to the sides, and two more pass through the diagonals of the square.

Ways to build a square

Based on the definitions, it seems that nothing is easier than building the right square. This is true, but on condition that you have all the measuring instruments. And if something is not available?

Let's look at the existing ways that will help us build this figure.

Measuring ruler and gon are the basic tools with which the square can be constructed most simply.

First, mark the point, let's say A, from it we will build the base of the square.

Using the ruler, set the distance equal to the length of the side from it to the right, say 30 mm, and place point B.

Now from both points, using a square, draw up perpendiculars of 30 mm each. At the ends of perpendiculars we put the points B and D, which we connect with each other, using the ruler - everything, the ABHG square with a side of 30 mm is ready!

Using a ruler and protractor it's also quite easy to build a square. Begin, as in the previous case from the point, say H, from it, postpone the horizontal segment, for example 50 mm. Set the point O.

Now connect the center of the protractor to the point H, mark the value of the angle 90 ⁰ , through it and the point H construct a vertical segment of 50 mm, at its end, put the point P. Then, in a similar way, construct the third segment from the point O through the angle 90 ⁰ , equal to 50 Mm, let it end with a point P. Connect the points of P and P. You have a square of the NORP with a side length of 50 mm.

You can construct a square using only the compass and ruler. If the size of the square is important to you and the length of the side is known, you will also need a calculator.

So, put the first point E - it will be it from the vertices of the square. Next, specify the location where the opposite vertex G is located, that is, keep the diagonal of the Hedgehog of your figure. If you are building a square in size, then having a side length, calculate the length of the diagonal according to the formula:

D = √2 * a, where a is the side length.

After you learn the length of the diagonal, draw a section of the EH of this value. From the point E, draw a semicircle with the help of a compass in the direction of the point Ж. Conversely, from the point Ж there is a semicircle in the direction of the point E, of radius 3E. Through the intersection points of these semicircles, using a ruler, construct a segment of 3I. HZ and ZI intersect at right angles and are the diagonals of the future square. By connecting the points EI, IZH, LZ and ZE with a ruler, you get the inscribed square EIZHZ.

Still there is an opportunity to construct a square with the help of one ruler. What is a square? This is a section of the plane bounded by intersecting segments (lines, rays). Therefore, we can construct a square with respect to the coordinates of its vertices. First, draw the coordinate axes. The sides of the square can lie on them, or the center of the intersection of the diagonals will coincide with the point of origin - this depends on your desire or the conditions of the problem. Perhaps your figure will be spaced from the axes at some distance. In any case, first mark two points by arbitrary values (arbitrarily or conditionally), then you will know the length of the side of the square. Now we can calculate the coordinates of the remaining two vertices, remembering that the sides of the square are equal and are pairwise parallel to each other. The last step is to connect all the points in series with each other using the ruler.

What are the squares?

A square is a well-defined figure and rigidly bounded by its definitions, therefore the types of squares are not diverse.

In non-Euclidean geometry, the square is perceived more widely - it is a quadrilateral with equal sides and angles, but the degree of angles is not given. This means that the angles can be 120 degrees ("convex" square) and, for example, 72 degrees ("concave" square).

If you ask what a square is, with a geometer or computer science, you will be told that it is a complete or planar graph (graphs from K ₁ to K ₄ ). And this is absolutely true. The graph has vertices and edges. When they get into an ordered pair, a graph is formed. The number of vertices is the order of the graph, the number of edges is its size. Thus, a square is a planar graph with four vertices and six edges, or K ₄ : 6.

Side of a square

One of the main conditions for the existence of a square - the presence of equal length sides - makes the side very important for various calculations. But at the same time it gives many ways for the length of the side of the square to be computed in the presence of very different initial data.

So, how to find the value of the side of the square?

• If you know only the length of the diagonal of the square d, then you can calculate the side by the following formula: a = d / √2.
• The diameter of the inscribed circle is equal to the side of the square and, consequently, to two radii, that is: a = D = 2R.
• The radius of the circumscribed circle can also help to calculate what the side of the square is equal to. We can determine the diameter D by the radius R, which in turn is equal to the diagonal of the square d, and we already know the formula for the side of the square through the diagonal: a = D / √2 = d / √2 = 2R / √2.
• From equality of sides it follows that one can know the side of a square (a) with its perimeter P or area S: a = √S = P / 4.
• If we know the length of the line that comes out of the corner of the square and crosses the middle of its adjacent side C, then we will also be able to know what the length of the side of the square is: a = 2C / √5.

That's how many ways there are to figure out such an important parameter as the length of the side of the square.

Volume of a square

The phrase itself is absurd. What is a square? It is a flat figure, having only two parameters - length and width. And the volume? This is a quantitative characteristic of the space that an object occupies, that is, it can be calculated only for three-dimensional bodies.

A three-dimensional body, with all its faces squares, is a cube. Despite the colossal and fundamental difference, schoolchildren quite often try to calculate the volume of a square. If someone succeeds, the Nobel Prize is guaranteed.

And to find out the volume of the cube V, it is enough to multiply all its three edges - a, b, c: V = a * b * c. And since by definition they are equal, the formula may look different: V = a ³ .

Values, parts and characteristics

A square, like any polygon, has vertices - these are the points at which its sides intersect. The vertices of the square lie on the circle circumscribed about it. A diagonal passes through the vertex to the center of the square, which is also a bisectrix and the radius of the circumscribed circle.

Since the square is a flat figure, it is impossible to cut and construct the cross-section of the square. But it can be the result of intersection of many volume bodies with a plane. For example, a cylinder. The axial section of the cylinder is a rectangle or a square. Even if you cross the body with a plane at an arbitrary angle, you can get a square!

But the square has one more relation to the section, and not to any, but to the Golden Section.

We all know that the Golden Ratio is a proportion in which one value refers to another as well as their sum to a larger value. In the generalized percentage expression, it looks like this: the original value (amount) is divided by 62 and 38 percent.

The golden section is very popular. It is used in design, architecture, and anywhere, even in the economy. But this is by no means the only proportion deduced by Pythagoras. There is, for example, another expression "√2". On its basis, dynamic rectangles are constructed, which, in turn, are the founders of the A (A6, A5, A4, etc.) formats. Why did it come about dynamic rectangles? Because their construction begins with a square.

Yes, first you need to build a square. Its side will be equal to the smaller side of the future rectangle. Then it is necessary to draw a diagonal of this square and, using the compass, the length of this diagonal should be postponed on the extension of the side of the square. From the point obtained at the intersection, we construct a rectangle, in which we again construct the diagonal and postpone its length on the extension of the side. If you continue using this scheme, you will get those same dynamic rectangles.

The ratio of the long side of the first rectangle to the short one will be 0.7. This is almost 0.68 in the Golden Section.

Angles of the square

Actually, something fresh to say about the corners is already difficult. All properties, they are the attributes of a square, we listed. As for the angles, there are four of them (as in every quadrangle), each corner in the square is a straight line, that is, it has a dimension of ninety degrees. By definition, there is only a rectangular square. If the angles of a larger or smaller size are already another shape.

Diagonals of a square are divided by its angles in half, that is, bisectors.

Equation of a square

If necessary, to calculate the value of different values of the square (area, perimeter, side lengths or diagonals) use different equations, which are derived from the properties of the square, basic laws and rules of geometry.

1. The square equation of a square

From the equations for calculating the area of quadrilaterals, we know that it (area) is equal to the product of length and width. And since the sides of the square are the same in length, its area will be equal to the length of any side erected in the second degree

S = a ² .

Using the Pythagorean theorem, we can calculate the area of a square, knowing the length of its diagonal.

S = d 2/2.

2. The equation of the perimeter of the square

The perimeter of a square, like all quadrilaterals, is equal to the sum of the lengths of its sides, and since they are all the same, we can say that the perimeter of the square is equal to the length of the side multiplied by four

P = a + a + a + a = 4a.

Again, the Pythagorean theorem will help us find the perimeter through the diagonal. It is necessary to multiply the length of the diagonal by two roots of two

P = 2√2d

3. The equation of the diagonal of the square

The diagonals of the square are equal, intersect at a right angle and are divided by the point of intersection in half.

They can be found from the above equations of area and perimeter of the square

D = √2 * a, d = √2S, d = P / 2√2

There are also ways to find out what the length of the diagonal of the square is. The radius of the inscribed circle is half its diagonal, hence

D = √2D = 2√2R, where D is the diameter and R is the radius of the inscribed circle.

Knowing the radius of the circumscribed circle, it is even easier to calculate the diagonal, because it is the diameter, that is, d = D = 2R.

It is also possible to calculate the length of the diagonal, knowing the length of the line emerging from the corner to the center of the side of the square C: d = √8 / 5 * C.

But do not forget that a square is a section of a plane bounded by four intersecting lines.

For the lines (and the figures formed by them), there are enough equations that do not need an additional description, but the line is infinite. And polygons are bounded by the intersection of lines. For them, you can use linear equations, combined into a system that defines straight lines. But it is necessary to specify additional parameters and conditions.

To determine polygons, it is necessary to compose an equation that would describe not a line, but a separate arbitrary segment, without the intervention of additional conditions and descriptions.

[X / x _i ] * [x _i / x] * y _i - this is the special equation for polygons.

The square brackets in it indicate the condition for excluding the fractional part of the number, that is, we must leave only an integer. Y _i is a function that runs in the parameter range from x to x _i .

Using this equation, we can derive new equations for the calculation of segments and lines consisting of several segments. It is basic, universal for polygons.

Remember that a square is a part of a plane, therefore its description of the type y = f (x) can be represented, more often than not, as a multivalued function, which in turn can be expressed through single-valued ones, if they are represented parametrically, that is, depending on Of any parameter t:

X = f (t), y = f (t).

So, if we use in the aggregate a universal equation and a parametric representation, then we can indeed derive an equation for the expression of polygons:

X = ((A2 + A3) * A5 + A4 * P) * Cos (L)

Y = ((A1 + A4) * A5 + A3 * P) * Sin (L),

Where

A1 = [1 / [T / P]] * [T / P]; A2 = [2 / [T / P]] * [[T / P] / 2]; A3 = [3 / [T / P]] * [[T / P] / 3]; A4 = [4 / [T / P]] * [[T / P] / 4]; A5 = TP * [T / P],

Where P is the diagonal of the rectangle, L is the slope angle to the horizontal of the diagonal P, T is the parameter varying in the range from P to 5P.

If L = 3,14 / 4, then the equation will describe the squares of different values, depending on the size of the diagonal P.

Applying a square

In the modern world, technology allows you to give different materials a square shape, more precisely a square section.

This is much more profitable, cheaper, more durable and safer. So, now make square pipes, piles, wire (wires) and even square threads.

The main advantages are obvious, they come out of elementary geometry. With the same size, the area of the inscribed circle is smaller than the square of the square into which it is inscribed, hence the throughput of a square tube or the energy capacity of a square wire will be higher than for round analogues.

Often, the materials of the square section are more aesthetic and convenient in use, installation, fastening.

When choosing these materials, it is important to correctly calculate the cross-section of the square so that the wire or pipe will withstand the required load. In each individual case, of course, such parameters as current or pressure, but without the basic geometric rules of the square, will not be necessary. Although the sizes of square sections are not so much calculated, how many are chosen according to the given parameters from the tables established by GOSTs for different industries.