A square is a quadrilateral in which all angles are straight, that is, equal to 90 degrees, all sides are equal, and opposite sides are parallel. The area of ​​a square is equal to the product of its two sides. To find the side of a square, if only its area is known, it is necessary to extract the square root of a number equal to the area of ​​the square, or find a number by multiplying which by the same number, we get a number equal to the area of ​​the square.

For example: the area of ​​a square is 25 cm2. The side is 5, because 5 * 5 = 25.

A rectangle is a geometric figure with opposite sides of equal length and parallel to each other. The sides of the rectangle intersect at right angles. Unlike a parallelogram, a rectangle has the same diagonals.

The main characteristic of a rectangle is its length and width.

A square is a special case of a rectangle and has similar characteristics. The difference between a square is in the same sides. The length and width of the square are the same.

The areas of a rectangle and a square have a similar method of determination, by multiplying the value of the length of the figure by its width.

Formula for the area of ​​a rectangle

• S pr. = A * b;
• S pr. - the area of ​​the rectangle;
• a - the length of the rectangle;
• в - the width of the rectangle.

Area of ​​a square formula

• S sq. = a * a = a 2;
• S sq. - square area;
• a - the numerical value of the length of the side of the square.

This means that to find the side of a square, it is necessary to extract the square root from the value of the area.

Find the side of the square

a = √ S sq. ;

For example, let's take the value of the area of ​​a square equal to 25 cm2 and find the value of the side of this figure.

a = √ 25 = 5 cm.

We check:

S = 5cm * 5cm = 25cm 2.

Answer: we find the side of the square by extracting square root from the value of the area.

A square is a rhombus with right angles. This figure is at the same time a parallelogram, rectangle and rhombus, possessing exceptional geometric properties. It is possible to detect the side of a square through its diagonal in several ways.

You will need

• - Pythagorean theorem;
• - ratio of angles and sides right triangle;
• - calculator.

Instructions

1. From what square diagonals are equal to each other (he inherited this quality "by inheritance" from the rectangle), then in order to discover side square it is enough to know the length of one diagonal. Diagonal and two sides square adjacent to it, represent rectangular (from the fact that all corners square straight) and isosceles (because all sides of this figure are equal) triangle. In this triangle the sides square are the legs, and the diagonal is the hypotenuse. To discover side square, use the Pythagorean theorem.

2. Since the sum of the squares of the legs, which are equal to a, is equal to the square of the hypotenuse, which we denote c (c? = A? + A?), Then the leg will be equal to the hypotenuse divided by the square root of the number 2, which also follows from the previous expression a = c /? 2. Let's say in order to discover side square with a diagonal of 12 cm, divide this number by the square root of 2. Get a = 12 /? 2 × 8.5 cm. Taking into account that the square root of 2 is not completely extracted, all results will have to be rounded with the required accuracy.

3. Side square find, using the ratio of angles and sides in a right-angled triangle, the one that is formed by the diagonal and the sides adjacent to it. Apparently, one of the angles of this triangle is a straight line (like the angle between the sides square), and the other two are equal to each other and amount to 45 ?. This quality stems from the isosceles of this triangle, from the fact that its legs are equal to each other.

4. To discover side square, multiply the diagonal by the sine or cosine of the angle 45? (they are equal to each other, as the adjacent and opposite legs sin (45?) = cos (45?) =? 2/2) a = c ?? 2/2. Let's say the diagonal is given square equal to 20 cm, it is necessary to detect it side... Calculate according to the above formula, the total will be the side square with the required degree of accuracy a = 20 ?? 2/2? 14.142 cm.

Often in geometric problems it is required to find the length of a side of a square if its other parameters are known, such as area, diagonal, or perimeter.

You will need

• Calculator

Instructions

1. If the area of ​​a square is famous, then in order to find the side of the square, you need to extract the square root of the numerical value of the area (because the area of ​​the square is equal to the square of its side): a =? S, where a is the length of the side of the square; S is the area of ​​the square. the side of a square will be the linear unit of length that corresponds to the unit of area. Say, if the area of ​​a square is given in square centimeters, then the length of its side will be easily obtained in centimeters. Example: The area of ​​a square is 9 square meters. Find the length of a side of a square. Solution: a =? 9 = 3 Answer: The side of a square is 3 meters.

2. In the case when the perimeter of the square is famous, to determine the length of the side, it is necessary to divide the numerical value of the perimeter by four (because the square has four sides of the same length): a = P / 4, where: a is the length of the side of the square; P is the perimeter of the square. The unit for the side of the square will be the same linear unit for length as for the perimeter. For example, if the perimeter of a square is given in centimeters, then the length of its side will also be in centimeters. Example: The perimeter of a square is 20 meters. Find the length of a side of a square. Solution: a = 20/4 = 5 Answer: The length of a side of a square is 5 meters.

3. If the length of the diagonal of a square is famous, up to the length of its side will be equal to the length of its diagonal divided by the square root of 2 (by the Pythagorean theorem, because the adjacent sides of the square and the diagonal make up a right-angled isosceles triangle): a = d /? 2 (because . a ^ 2 + a ^ 2 = d ^ 2), where: a is the length of the side of the square; d is the length of the diagonal of the square. The unit of measure for the side of the square will be the unit of length that is the same as for the diagonal. Say, if the diagonal of a square is measured in centimeters, then the length of its side will be in centimeters. Example: The diagonal of a square is 10 meters. Find the length of the side of the square. Solution: a = 10 /? 2, or approximately: 7.071 Answer: The length of the side of the square is equal to 10 /? 2, or approximately 1.071 meters.

Representation " ratio”Is used in different areas skills. Social sciencies comprehend ratio the number of men and women employed and unemployed. The economy is famous ratio needs and suggestions. Business evaluates ratio debt and equity capital. In mathematics, research ratio sides of geometric shapes. It's fun in sports ratio height and weight of a person. In a general sense ratio Is an indicator that allows you to evaluate the difference between values. As an illustration, we find ratio the distance traveled and the remaining path of a car moving from one city to another.

Instructions

1. Determine the path traveled by the car. It is possible that the car has covered 120 km.

2. Determine the path that remains for the car to pass. Let it be 100 km left to go.

3. Discover ratio between the traversed and the remaining path. We divide 120 km by 100 km, we get 1.2.

4. Make a summary. Because the ratio is greater than 1, the car has traveled more than half the required distance. The distance covered is 1.2 times the remaining distance.

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Note!
In order for the driver of the car to have enough water and food for each path, he must monitor the ratio between the food eaten and the remaining food. This ratio should correspond to the ratio between the distance traveled and the remaining path.

Useful advice
For various phenomena of life, scientists have developed the desired ratios to which one must gravitate for the greatest efficiency of the process. Such desirable ratios are tabulated in reference books. Let's say the desired ratio of height and weight of a person is famous. Deviation from these values ​​in one direction or another shows the experts what is allowed to improve. If you wanted to evaluate whether it is cool or shoddy - the ratio of 1.2 for a car, you would have to refer to the appropriate reference book. In utilitarian activity, people, on their own skill, learn which ratios are cool, even if they are not indicated in any tables. For example, if the journey from one city to another takes 2 days, the driver of the car by skill may know that on the 1st day it is necessary to drive more than on the 2nd, since fatigue accumulates. And the higher the ratio, the better. He can drive so that at the end of the 1st day the ratio of the distance covered to the remaining path is at least 1.8. Tracking can tell you the best ratio for all the affairs of your life.

The square is a lovely and simple flat geometric shape. It is a rectangle with equal sides. How to detect diagonal square, if the length of its side is famous?

Instructions

1. Diagonal square find quite primitively, using the Pythagorean theorem. Let us divide the square diagonal yu into two equal triangles. In this case diagonal will be the hypotenuse of one of the triangles. And, as it is known, the square of the hypotenuse is equal to the sum of the squares of the legs. that the legs are the sides square and they are equal, the formula for calculating the diagonal square on its side is hefty primitive: the length of the diagonal square equal to the length of its side multiplied by the root of 2.

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Useful advice
If the accuracy of the mathematical total is not very important, then instead of the root of 2, it is allowed to use its approximate value of 1.41.

Tip 5: How to find the side of a square if its diagonal is known

The square is one of the most primitive geometric shapes in terms of calculating its parameters - the lengths of the sides and diagonals, area and perimeter. This is determined by the fact that, unlike other polygons, the values ​​of all its angles are invariably famous, and it is also enough to know the length of each one side. Finding the side length square along the famous length of the diagonal, both in general terms and with actual calculations is not difficult.

Instructions

1. Use the Pythagorean theorem, the algebraic formulation of which states that in a right-angled triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse: a? + b? = c ?. Because the diagonals square divide it into two just such right-angled triangles, in which, moreover, the lengths of the legs are identical, then it is allowed to formulate such a quality square, how geometric shape: the square of the length of the diagonal is equal to twice the square of the length of the side (2a? = c?). This implies that the length of the side is equal to the square root of half square the length of the diagonal: a = √ (c? / 2).

2. Use Google's built-in calculator to actually calculate side length square... For example, if the famous diagonal length is 15 centimeters, then going to the search engine site, enter the following query: "root of ((15 squared) / 2)". If you are used to using the ^ symbol to denote exponentiation and sqrt to denote the square root operation, then Google correctly understands this query: "sqrt (15 ^ 2/2)". In any case, the result will be identical: side length square equal to 10.6066017 centimeters.

3. Use, say, a software calculator from the standard Windows operating system as alternative method to calculate the side length square... The link to its launch is hidden quite large in the main menu of the system - after clicking on the "Start" button, you must open the "All programs" section, go to the "Typical" subsection, click the "Utilities" section and prefer the "Calculator" item. More swift method - press the key combination WIN + R, enter the command calc and press the Enter key.

4. Enter the known side length, then press the star key and Enter - this will perform the squaring operation. After that press the forward slash key, type two and press Enter. Later on, click on the button labeled sqrt and you will see the desired side length square- 10.606601717798212866012665431573 centimeters.

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Square Is a regular quadrangle 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 a diagonal.
To calculate the area, consider the formula for the area of ​​a square through the 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 formula for the area of ​​a right-angled triangle or by the Pythagorean theorem.

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

For 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 quadrangle 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 diameter, now we will try to find the length of the side:
Let's substitute the value into the expression:
The side of the square will be 2.1 cm long.

It is very simple 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.

You are 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 the circle is equal to the diagonal of the square. We substitute the value into the formula for calculating the area of ​​a square through its diagonals:

The area of ​​the square is 18

Area of ​​a square through the perimeter

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