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 big square the same square by π/2 times.

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

Quite often, this figure is considered as a 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 squared 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.

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 Square root area value numbers (since the area of ​​a square is equal to the square of its side): z = 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: Given the area of ​​a square 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.