Square is a regular quadrilateral in which all angles and sides are equal to each other.

Quite often this figure is considered as special case or . The diagonals of a square are equal to each other and are used in the formula for the area of ​​a square through the diagonal.
To calculate the area, consider the formula for the area of ​​a square in terms of diagonals:

That is, the area of ​​a square is equal to the square of the length of the diagonal divided by two. Given that the sides of the figure are equal, you can calculate the length of the diagonal from the area formula of a right triangle or using the Pythagorean theorem.

Consider an example of calculating the area of ​​a square through the diagonal. Let a square with a diagonal d = 3 cm be given. It is necessary to calculate its area:

Using this example of calculating the area of ​​a square through the diagonals, we got the result 4.5 .

Square area across side

You can also find the area of ​​a regular quadrilateral by its side. The formula for the area of ​​a square is very simple:

Since in the previous example of calculating the area of ​​a square, we calculated the value by the diameter, now let's try to find the length of the side:
Substitute the value in the expression:
The length of the side of the square will be 2.1 cm.

It is very easy to use the formula for the area of ​​a square inscribed in a circle.

The diameter of the circumscribed circle will be equal to the diameter of the square. Since a square is considered a regular rhombus, you can use the formula for calculating the area of ​​a rhombus. It is equal to half the product of its diagonals. The diagonals of the square are equal, so the formula will look like this:
Consider an example of calculating the area of ​​a square inscribed in a circle.

Given a square inscribed in a circle. The diagonal of the circle is d = 6 cm. Find the area of ​​the square.
We remember that the diagonal of a circle is equal to the diagonal of a square. We substitute the value in the formula for calculating the area of ​​​​a square through its diagonals:

The area of ​​the square is 18

Square area through perimeter

In some problems, the perimeter of the square is given by the conditions and the calculation of its area is required. The formula for the area of ​​a square through the perimeter is derived from the value of the perimeter. Perimeter is the sum of the lengths of all sides of the figure. Because in a square of 4 equal sides, then it will be equal From here we find the side of the figure The area of ​​\u200b\u200bthe square according to the usual formula is considered as follows:.
Consider an example of calculating the area of ​​a square through the perimeter.

When they have the same lengths of diagonals, sides and equal angles.

Square properties.

All 4 sides of a square have the same length, i.e. the sides of the square are:

AB=BC=CD=AD

Opposite sides of a square are parallel:

AB|| CD, BC|| AD

All diagonals divide the corner of the square into two equal parts, so they turn out to be the bisectors of the corners of the square:

∆ABC = ∆ADC = ∆BAD = ∆BCD

∠ ACB=∠ ACD=∠ BDC=∠ BDA=∠ CAB=∠ CAD=∠ DBC=∠ DBA = 45°

The diagonals divide the square into 4 identical triangles, in addition, the triangles obtained at the same time are both isosceles and rectangular:

∆AOB = ∆BOC = ∆COD = ∆DOA

The diagonal of a square.

Diagonal of a square is any segment that connects the 2 vertices of the opposite corners of the square.

The diagonal of any square is √2 times the side of this square.

Formulas for determining the length of the diagonal of a square:

1. The formula for the diagonal of a square in terms of the side of a square:

2. The formula of the diagonal of a square in terms of the area of ​​a square:

3. The formula of the diagonal of a square in terms of the perimeter of a square:

4. The sum of the angles of a square = 360°:

5. Diagonals of a square of the same length:

6. All diagonals of the square divide the square into 2 identical figures that are symmetrical:

7. The angle of intersection of the diagonals of the square is 90 °, crossing each other, the diagonals are divided into two equal parts:

8. The formula for the diagonal of a square in terms of the length of the segment l:

9. The formula for the diagonal of a square in terms of the radius of the inscribed circle:

R- radius of the inscribed circle;

D- diameter of the inscribed circle;

d is the diagonal of the square.

10. The formula for the diagonal of a square in terms of the radius of the circumscribed circle:

R- radius of the circumscribed circle;

D- diameter of the circumscribed circle;

d- diagonal.

11. The formula for the diagonal of a square through a line that comes out of the corner to the middle of the side of the square:

C- a line that goes from the corner to the middle of the side of the square;

d- diagonal.

Inscribed circle in a square- this is a circle adjacent to the midpoints of the sides of the square and having a center at the intersection of the diagonals of the square.

Inscribed circle radius- side of the square (half).

Area of ​​a circle inscribed in a square less than the area of ​​a square by π/4 times.

Circle circumscribed around a square is a circle that passes through 4 vertices of the square and which has a center at the intersection of the diagonals of the square.

Radius of a circle inscribed around square greater than the radius of the inscribed circle by √2 times.

Radius of a circle inscribed around a square equals 1/2 of the diagonal.

Area of ​​a circle circumscribed around a square the larger area of ​​the same square is π/2 times.

Often, in the course of planning a summer cottage landscape, it becomes necessary to “squeeze” some building, for example, a barn, a bathhouse or a gazebo, into a certain fragment of the site. At the same time, it is necessary to very accurately determine the geometric dimensions of the future structure, since in case of an error during construction, one will have to face big problems. A similar problem also arises when planning the interior space of a residential building. Therefore, it will be useful to know how to calculate the side of a square, knowing other characteristics of a geometric figure - area, diagonal, perimeter.

How to find the side of a square if only its area is known?

The easiest way to calculate the size of a square is if its area is known. Such a need is often encountered when building or laying out a garden. For example, if you need to determine the size of the future greenhouse, which should occupy a certain number of square meters. Similar calculations have to be carried out also when it is necessary to distinguish single space first or second floor, highlighting in it a square room for a bedroom, kitchen, living room or, in the end, a bathroom. At the same time, there are building codes, according to which the area of ​​​​functional premises should not be less than certain values.

As you know, the area of ​​a rectangle is determined by multiplying its sides. A square is a regular rectangle, the sides of which are equal, therefore, to calculate its area, one of the sides must be raised to the second power. Thus, to find the side of a square with a known area, you need to extract the square root from it. For example, if it is planned to build a square building with an area of ​​​​16 square meters. m., then each side should be 4 m. If the solution of the square root does not fit into an integer value (for example, the area is 17.5 sq. m.), then you can use a regular calculator to calculate. It is available in modern cell phones or among applications operating system Windows.

How to find the side of a square if the perimeter is known?

Such a task may be faced by a summer resident, for example, when determining the size of a greenhouse or greenhouse. The perimeter in such cases is determined based on the number of available building materials. If at the same time it is incorrect to set the side of the structure, then you will definitely run into problems. If the dimensions are too small, this will entail a loss of usable area. And if you plan too much great importance On the other hand, there will not be enough materials, you will have to buy them in addition, and these are additional costs and troubles.

When the perimeter of the square is known, then to calculate the length of its side, it is enough to divide the numerical value of the perimeter by the number of sides, that is, by 4. For example, a gardener has 40 m of a metal corner at his disposal, which is used during the construction of a greenhouse as a frame for attaching plastic sheets. Then you need to divide this number by 2, because there will be two guides - above and below. Thus, the perimeter of the future greenhouse is 20 m, which means that its side should be 5 m.

How to find the side of a square if only the diagonal is known?

This is the most difficult option, although in this case the calculations are not particularly difficult. The Pythagorean theorem comes to the rescue here, according to which, the squared hypotenuse is equal to the sum of the legs, also squared. Moreover, the diagonal of a square with two adjoining sides is nothing but a right triangle. Moreover, since the sides are equal, the figure is still isosceles. And this means that the Pythagorean formula acquires a different formulation: the diagonal raised to the second power turns out to be equal to the square of the side multiplied by 2. It follows that to determine the side of the square, you need to first raise its diagonal to the second power, then divide by 2, and after calculate the square root of this value.

For example, if the diagonal of the proposed structure is planned to be 10 m, then, raising it to the second power, we get 100, divide by 2 and calculate from the result Square root. As a result, the side of the square is determined as 7.07 m.

Useful advice

For practical calculations of the length of the sides of a square, you can use the help of tools such as the calculator built into the Google search engine. To do this, just enter the specified site and enter the following inscription in the search field: "root of ((D squared) / 2)". Instead of the symbol "D", of course, you need to substitute the value of the length of the diagonal. By the way, Google allows the use of the characters ^ or sqrt to denote exponentiation or root calculations, respectively. So, if it is more convenient for someone, then you can replace the previous expression with the entry: “sqrt (D ^ 2/2)”.

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Often in geometry it is necessary to find the length of the side of a square, while its parameters are known: perimeter, area, length of the diagonal.

A square is a rhombus or rectangle whose sides are equal to each other. The corners of the square are also equal to each other and have 90 ° each. Consider how to find the side of a square given one of the above parameters.

Finding the side of a square by its perimeter

In this case, to find the length of the side of the square, it is necessary to divide the value of the perimeter of the square by 4 (since the square has 4 sides equal to each other): z \u003d P / 4, where z is the length of the side of the square; P is the perimeter of the square.

The unit of measure for one side of the square will be the same unit for length as its perimeter. For example, if the perimeter of a square is given in millimeters, then the length of its side will also be in millimeters.

For example: The perimeter of a square is 40 meters. When solving this problem, we get: z \u003d 40/4 \u003d 10. The length of the side of the square is 10 meters.

Finding the side of a square given its area

In this case, to find the length of the side, you need to get the square root of the area value (since the area of ​​​​the square is equal to the square of its side): z \u003d vS, where z is the length of the side of the square; S is the area of ​​the square.

The unit for one side of a square will be the same unit for length as its area. For example, if the area of ​​a square is given in square millimeters, the length of its side will simply be in millimeters.

For example: The area of ​​a square is 16 square meters. When solving this problem, we get: z = v9 = 3. The length of the side of the square is 4 meters.

Finding the side of a square from its diagonal

In this case, the length of the side of the square will be equal to the length of the diagonal of the square divided by the square root of 2 (for the Pythagorean theorem, since the adjacent sides of the square and its diagonal form an isosceles right triangle). To find the side of a square diagonally, you need: z \u003d d / v2 (since z 2 + z 2 \u003d d 2), where: z is the length of the side of the square; d is the length of the diagonal of the square.

The unit for one side of the square will be the same length unit as its diagonal. For example, if the diagonal of a square is given in millimeters, then the length of its side will also be in millimeters.

For example: Given a square diagonal of 20 meters. When solving this problem, we get: z = 20/v2, which is approximately equal to 20/1.4142. The length of the side of the square is 20/v2 meters, or approximately 14.142 meters.

Now you know how to find the side length of a square given its perimeter, area, or diagonal length.