answer questions about what a square, can be set.It all depends on whom you addressed this issue.The musician says that the square - a 4, 8, 16, 32 bars or jazz improvisation.The baby - it's a game with a ball or a children's magazine.The printer will send you to study font size and technique - metal-profile species.
Many other values of this word, but today we ask the math.So ...
Understand this figure we will gradually, from simple to complex, and start with the history of the square.As it appeared, as perceived by people, scientists from different countries and civilizations?
history study square
ancient world perceives the square, mainly the four corners of the world.In general, despite the many quads, just at the main square of the number - four.For the Assyrians and the Peruvian square - the whole world, that is, it represents the four main directions of the compass.
Even the universe is like a square, also divided into four parts - the vision of North America.For the Celts, the universe - it is three square, nested, and the center follow the four (!) Of the river.And all the Egyptians worshiped this figure!
first described by the square of the mathematical formulas Greeks.But for them, this polygon has only negative characteristics.Pythagoras did not like even numbers, seeing them as weak and feminine.
even in religion there is a square.In Islam, the Kaaba - the navel of the earth - is not some kind of spherical, namely a cubic shape.
In India, the main grapheme representing the earth or the earth symbol, the square was a crossover.Again, we are talking about the four cardinal points, the four regions of the earth.
In China, the square - is peace, harmony and order.Chaos is vanquished building square Vary.A square inscribed in a circle, is the basis for the vision of the world, symbolizing the unity and connection of Space and Earth.
Pagan Russia - Square Svarog.This symbol is also called Svarog Star or Star of Russia.It is rather complicated, as composed of intersecting lines and closed.Svarog - god-smith, the main creator, the creator of heaven itself, and in the representation of Rus.This symbol is a rhombus, which again speaks of the Earth and its four directions.And the star with four rays - 4 directions, 4 faces of Svarog - his omniscience.A cross beams - fireplace.
Interesting facts about the square
most popular phrase that comes to mind about our main character - "Black Square".
Malevich painting is still very popular.The author after its creation have long suffered 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 closely, you will notice that the plane of the square is not smooth, and black paint in the cracks there are many different color shades.Apparently, in the beginning there was a certain composition, which the author did not like, and he closed it from our eyes to this figure.Black Square as anything - a black hole, only the magic square shape.A void, as it is known, attracts ...
Another very popular "magic squares".In fact it is - Table naturally square filled numbers in each box.The sum of these numbers is the same in all rows, columns and diagonals (individually).If diagonal excluded from the equation, the square - semimagic.
Albrecht Durer in 1514 created the painting "Melancholia I", which depicted the 4x4 magic square.It sum of the numbers of columns, rows, diagonals, and even the interior of the square is equal to thirty-four.
On the basis of these tables were very interesting and popular puzzle - "Sudoku".
Egyptians were the first to carry out interconnection line numbers (date of birth) and character traits, abilities and talents of the person.Pythagoras took this knowledge, few processed and placed in the square.The result was a square of Pythagoras.
This is a separate area in numerology.From the date of birth of a person by adding the four basic calculate numbers that are placed in Pythagoras Square (square).And laid out all the secret information on your energy, health, talent, luck, temperament and other things on the shelves.On average, the accuracy of the polls is 60% -80%.
What is a square?
square is called a geometric figure.A square form - a quadrangle having equal sides and corners.More precisely, the rectangle is called correct.
The square has its own characteristics.It:
- sides of equal length;
- equal angles to each other - straight (90 degrees).
Because of these characteristics and features of the squared circle can be inscribed, and describe it around.The circumscribed circle will apply to all of its vertices inscribed - the middle of its sides.Their focus will coincide with the center of the square, and share all of its diagonally in half.The latter, in turn, is equal, and the angles of a square divided into equal parts.
One diagonal divides the square into two isosceles triangles, the two - to four.
Thus, if the length of a side of a square - t, the length of the radius of the circumscribed circle of - R, and an inscribed - r, then
- square footprint, or the area of a square (S) is equal to S = t2 = 2R2 = 4r2;
- square perimeter P should be calculated according to the formula P = 4t = 4√2R = 8r;
- length of the radius of the circle R = (√2 / 2) t;
- inscribed - r = t / 2.
Footprint square can still be calculated if his side (a) or the length of its diagonals (c), then the formula would look like, respectively: S = a2 and S = 1 / 2c2.
What is the square, we found you.Let's take a closer look at the details, because the figure of the square is the symmetrical rectangle.It has five axes of symmetry, one (fourth order) passes through the center and is perpendicular to the plane of the square, and four others - twofold symmetry axis, two of which are parallel to the sides, and two passes through the diagonal of a square.
way of constructing a square
based on the definition, it seems that there is nothing easier than to build a perfect square.This is true, but on the condition that you have all the measurement tools.And if something is not available?
Let's look at the existing methods, which will help us build this figure.
measuring ruler and square - are the main tools through which you can most easily build a square.
first check point, say A, from it we construct the base of a square.
Using a ruler aside from it to the right a distance equal to the length of the part, for example 30 mm, and set point B.
now on both points, using a square, draw up perpendiculars of 30 mm each.At the ends perpendicular to put points C and D, which are connected to each other using a ruler - all square ABCD with the side 30 mm ready!
with a ruler and protractor is also quite easy to build a square.Start, as in the previous case in terms, for example N, aside from its horizontal interval, for example 50 mm.Put the point O.
Now connect the center of the protractor to the H-point, place a mark at the angle of 900, and through it the H-point, build a vertical line 50 mm at its end with a point P. Further, in this way build a third length from the point O by angle900, equal to 50 mm, even if it ends up point P. Connect the dots R and R. You have turned OGMF square with a side length of 50 mm.
can build a square, using only a compass and straightedge.If you have important size of the square and is known for the length of the side, and it will take a calculator.
So, put the first point of E - it will be it of the vertices of a square.Next, specify the location where it will be located opposite the top of F, that is, wait diagonal HEDGEHOG your figure.If you build a square in size, with the length of the side, calculate the length of the diagonal of the formula:
d = √2 * a, where a - length of the side.
Once you know the length of the diagonal length of hedgehog build this value.From point E with a compass in the direction of the point F spend a half-circle radius hedgehog.Conversely, from the point F - semicircle towards point E, radius SAME.Through the point of intersection of these semicircles, using a ruler, draw a segment link.HEDGEHOG GI and intersect at right angles and diagonals are the future of a square.Connect the dots Eu, IL, ZHZ and WE with a ruler, you will receive an inscribed square EIZHZ.
It is still possible to construct a square with one line.What is Square?This section of the plane bounded by intersecting segments (lines, rays).Therefore, we can build a square in the coordinates of its vertices.First, draw the axes.Side of the square can lie to them, or the center of the intersection of the diagonals will coincide with the point of origin - it depends on your desire or problem conditions.Perhaps your figure will be spaced from the axis at a distance.In any case, the first mark on the numeric values (randomly or conditionally) two points, then you will be known for the length of the side of the square.Now you can calculate the coordinates of the remaining vertices of the two, remembering that the sides of the square are equal and parallel to each other in pairs.The final step - to connect all points sequentially interconnected with a ruler.
What are the boxes?
Square - a figure clearly defined and strictly limited their definitions, so the kinds of squares do not differ diversity.
in non-Euclidean geometry of the square is seen more broadly - a rectangle with equal sides and angles, but the degree of angle is not specified.This means that the angles may be 120 degrees ("convex" square), for example, by 72 degrees ("concave" square).
If you ask what the square from geometry or computer, you will answer that - it is a complete or planar graph (columns K1 K4).And it's absolutely true.The count has vertices and edges.When they get up in the ordered pair, form a graph.The number of vertices - this is the order of the graph, the number of edges - its size.Thus, a square - a planar graph with four vertices and six fins or K4: 6.
side of the square is one of the main conditions for the existence of a square - the presence of equal length sides - makes a very important aside for various calculations.But at the same time provides many ways to a square side length was calculated in the presence of various basic data.
So, how to find the value of the square?
- If you know only the length of the diagonal of the square d, then you can calculate the direction of the following formula: a = d / √2.
- diameter of the inscribed circle is the side of a square, and hence two radii, that is: a = D = 2R.
- radius of the circle can also help figure out what is the side of the square.We may identify the radius R of the diameter D, which, in turn, is equal to the diagonal of a square d, and the formula for the diagonal of the square through we know: a = D / √2 = d / √2 = 2R / √2.
- From equality of arms implies that the know side of the square (a) can by using its perimeter P or the area S: a = √S = P / 4.
- If we know the length of the line that goes from the corner of the square and intersects the middle of its adjacent side C, then we will also be able to find out what is the length of the side of the square: a = 2C / √5.
That's how many ways there are to see such an important parameter as the length of the square.
volume square
very phrase is absurd.What is Square?This is a plane figure having only two parameters - the length and width.And the volume?This is a quantitative characteristic of the space occupied by the object, that is, it can be calculated only for the bulk of bodies.
volumetric body, all of whose faces are squares - the cube.Despite the enormous and fundamental difference, students often try to calculate the volume of a square.If someone does succeed, the Nobel Prize is provided.
And to find out the volume of a cube V, is sufficient to multiply all three of his ribs - a, b, c: V = a * b * c.And since they are by definition equal, the formula may look different: V = a3.
quantities and characteristics of the
The square, like any polygon vertices have - is the point at which the cross on his part.The tops of the square lie on a circle described around him.Through the top of the center of the square diagonal passes, which is also the bisector and the radius of the circle.
Since the square - a plane figure, then cut and build a cross-section of the square is impossible.But it may be the result of crossing many bulky body plane.For example, a cylinder.The axial cross-section of a cylinder - a rectangle or square.Even when crossing the plane of the body at an arbitrary angle can turn a square!
But the square has another attitude to the cross section, but not to some, but to the golden section.
We all know that the Golden Ratio - a ratio in which one value belongs to another as well as their sum to a larger size.In summary, this percentage is as follows: initial value (amount) is divided by 62 and 38 percent.
golden section is very popular.It is used in the design, architecture, yes anywhere, even in the economy.But it is not only the proportion derived by Pythagoras.For example, there is still the expression "√2".On this basis we construct dynamic rectangles, which, in turn, is the founder of the group A of formats (A6, A5, A4, etc.).Why we are talking about the dynamic rectangles?Because their construction begins with a square.
Yes, first you need to build a square.His side will be equal to the lesser of the next rectangle.Then you need to hold the diagonal of the square and using the compass, the length of the diagonal to postpone the continuation of the square.From the resulting points are building at the intersection of the rectangle, which again build the diagonal length and lay it on the extended hand.If you continue to work on this scheme, will receive the very dynamic rectangles.
ratio of the long side of the first rectangle to be short 0.7.It's almost 0.68 in the Gold section.
angles square
Actually, something fresh to say about the angles is difficult.All the properties, they are also signs of the square, we have listed.As for the corners, four of them (as in any quadrangle), each corner of the square - a line that is has a size of ninety degrees.By definition, there is a rectangular square.If the corners of the larger or smaller - this is a different figure.
diagonal of the square corners divide it in half, then there are the bisectors.
Equation square
If necessary, calculate the value of the different variables at the square (area, perimeter, length of sides and diagonals) use various equations that are derived from the properties of the square, the main laws and regulations geometry.
1. Equation square square
From the equations to calculate the area of rectangles, we know that it (the area) is the product of length and width.And as the square side equal in length, the area it will be equal to the length of either side, built in the second degree
S = a2.
Using the Pythagorean theorem, we can calculate the area of a square, knowing the length of the diagonal.
S = d2 / 2.
2. Equation
perimeter of the square perimeter of the square, as well as all quads, 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 Pythagorean theorem will help us find the perimeter through a diagonal.It is necessary to multiply the length of the diagonal on the two roots of the two
P = 2√2d
3. The equation diagonal square
diagonal of the square are intersect at right angles and divide the intersection of two.
You can find them on the basis of the above equations of area and perimeter of a square
d = √2 * a, d = √2S, d = P / 2√2
there are still ways to find out what is the length of the diagonal of the square.The radius of a circle inscribed in a square is equal to half of its diagonal, hence
d = √2D = 2√2R, where D - diameter, and R - radius of the inscribed circle.
Knowing the radius of the circle, calculate the diagonal is even easier, because it is the diameter, that is, d = D = 2R.
is also possible to calculate the length of the diagonal, knowing the length of the line coming out of the corner to the center of the square C: d = √8 / 5 * C.
But do not forget that the square - a section plane defined by four intersecting lines.
