How to find the sum of the perimeters of squares. Perimeter, area and volume

Square is a geometric figure, which is a quadrilateral with all angles and sides equal. It can also be called rectangle, whose adjacent sides are equal, or rhombus where all angles are equal 90º. Thanks to the absolute symmetry find square or perimeter of the square very easy.

Instruction:

• First, let's define that perimeter is called the sum of the lengths of all sides of a flat geometric figure, which is measured in the same quantities as the length. There are two ways to calculate the perimeter of a square.

Through the length of the side and diagonal

• Insofar as perimeter of the square is determined by the sum of the lengths of all its sides, and the sides of this figure are equal, then you can calculate the value of this value by multiplying the length of one side by the number " 4 ". Accordingly, the formulas will look like this: P = a + a + a + a or P = a * 4 , where R- it perimeter of the square and a – side length.
• In addition, depending on the condition of the problem, the perimeter of a square can be calculated by multiplying the length of its diagonal by two roots of two: P \u003d 2√2 * d , where R- it perimeter of the square and d- his diagonal.
• Some tasks require finding perimeter of the square knowing him square . It won't be difficult to do this either. The area of ​​a given figure is equal to the length of its side squared: S = a 2 , where S – square area and a –the length of its side. Or the area is equal to the square value of the length of its diagonal, divided by two: S = d2/2 , where S- still the same square and d – square diagonal.
• Knowing the formulas and the value of the area, it is not difficult to find the length of the side or the length of the diagonal, and then return to the formulas for calculating the perimeter and calculate its value.

Through the radius of the inscribed and circumscribed circle

• Finally, it is important to understand and how to find perimeter of the square if known circle radius described around it (or, on the contrary, inscribed in it). A circle inscribed in a given geometric figure touches the middle of each side, and its radius is equal to half of any side: R in \u003d ½ a , where R in – inscribed circle radius and a – side of a square.
• Circumscribed circle passes through all the vertices of the square and its radius is equal to half the length of the diagonal: R o \u003d ½ d , where R o - this radius of a circle circumscribed about a square and d- his diagonal.
• Therefore, in the first case, the perimeter will be calculated by the formula: R = 8 R in , and in the second: P = 4 x √2 x R o .

Using websites and an online calculator

• If you suddenly for some reason forgot the formulas, then the Internet will help refresh your knowledge. Go to the browser, open the search engine page and type in the appropriate query in the window, for example: " square perimeter formula". The system will give a huge number sites reference character, which will help you in this matter, as well as allow you to cope with solving problems related to other geometric shapes.
• In addition, if you do not want to understand the formulas and calculate the values ​​yourself, then you can use the services online calculators . An example is a website. Chapter " Formulas for the perimeter of geometric shapes» contains theoretical information supported by visual illustrations. If you follow the link " online calculator ”, which is located in the window of each figure, then a page for calculations will open in front of you.
• Select in the box below what you are going to calculate based on perimeter of the square(side or diagonal), and then enter the available data. The system will issue result , guided by the established formulas.
• In addition, on the site you will find a lot of other information that can make it easier to work with math problems. If you wish, you can search for more convenient or informative reference sites.
• If you cannot figure out the very course of solving the problem, then here you can ask for help from people who are well versed in the methodology for solving mathematical exercises. They can always be found on the corresponding forums , for example, or.

Lesson and presentation on the topic: "Perimeter and area of ​​a rectangle"

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What is a rectangle and a square

Rectangle is a quadrilateral with all right angles. So the opposite sides are equal to each other.

Square is a rectangle with equal sides and angles. It is called a regular quadrilateral.

Quadrilaterals, including rectangles and squares, are denoted by 4 letters - vertices. Latin letters are used to designate vertices: A, B, C, D...

Example.

It reads like this: quadrilateral ABCD; square EFGH.

What is the perimeter of a rectangle? Formula for calculating the perimeter

Perimeter of a rectangle is the sum of the lengths of all sides of the rectangle, or the sum of the length and width multiplied by 2.

The perimeter is indicated by the Latin letter P. Since the perimeter is the length of all sides of the rectangle, the perimeter is written in units of length: mm, cm, m, dm, km.

For example, the perimeter of a rectangle ABCD is denoted as P ABCD, where A, B, C, D are the vertices of the rectangle.

Let's write the formula for the perimeter of quadrilateral ABCD:

P ABCD = AB + BC + CD + AD = 2 * AB + 2 * BC = 2 * (AB + BC)

Example.
A rectangle ABCD is given with sides: AB=CD=5 cm and AD=BC=3 cm.
Let's define P ABCD .

Solution:
1. Let's draw a rectangle ABCD with initial data.
2. Let's write a formula for calculating the perimeter of this rectangle:

P ABCD = 2 * (AB + BC)

P ABCD=2*(5cm+3cm)=2*8cm=16cm

Answer: P ABCD = 16 cm.

The formula for calculating the perimeter of a square

We have a formula for finding the perimeter of a rectangle.

P ABCD=2*(AB+BC)

Let's use it to find the perimeter of a square. Considering that all sides of the square are equal, we get:

P ABCD=4*AB

Example.
Given a square ABCD with a side equal to 6 cm. Determine the perimeter of the square.

Solution.
1. Draw a square ABCD with the original data.

2. Recall the formula for calculating the perimeter of a square:

P ABCD=4*AB

3. Substitute our data into the formula:

P ABCD=4*6cm=24cm

Answer: P ABCD = 24 cm.

Problems for finding the perimeter of a rectangle

1. Measure the width and length of the rectangles. Determine their perimeter.

2. Draw a rectangle ABCD with sides 4 cm and 6 cm. Determine the perimeter of the rectangle.

3. Draw a CEOM square with a side of 5 cm. Determine the perimeter of the square.

Where is the calculation of the perimeter of a rectangle used?

1. A piece of land is given, it needs to be surrounded by a fence. How long will the fence be?

In this task, it is necessary to accurately calculate the perimeter of the site so as not to buy extra material for building a fence.

2. Parents decided to make repairs in the children's room. You need to know the perimeter of the room and its area in order to correctly calculate the number of wallpapers.
Determine the length and width of the room you live in. Determine the perimeter of your room.

What is the area of ​​a rectangle?

Square- This is a numerical characteristic of the figure. Area measured square units lengths: cm 2, m 2, dm 2, etc. (centimeter squared, meter squared, decimeter squared, etc.)
In calculations, it is denoted by the Latin letter S.

To find the area of ​​a rectangle, multiply the length of the rectangle by its width.
The area of ​​the rectangle is calculated by multiplying the length of AK by the width of KM. Let's write this as a formula.

S AKMO=AK*KM

Example.
What is the area of ​​rectangle AKMO if its sides are 7 cm and 2 cm?

S AKMO \u003d AK * KM \u003d 7 cm * 2 cm \u003d 14 cm 2.

Answer: 14 cm 2.

The formula for calculating the area of ​​a square

The area of ​​a square can be determined by multiplying the side by itself.

Example.
V this example the area of ​​a square is calculated by multiplying side AB by width BC, but since they are equal, side AB is multiplied by AB.

S ABCO = AB * BC = AB * AB

Example.
Find the area of ​​the square AKMO with a side of 8 cm.

S AKMO = AK * KM = 8 cm * 8 cm = 64 cm 2

Answer: 64 cm 2.

Problems to find the area of ​​a rectangle and a square

1. A rectangle with sides of 20 mm and 60 mm is given. Calculate its area. Write your answer in square centimeters.

2. A suburban area was bought with a size of 20 m by 30 m. Determine the area of ​​\u200b\u200bthe summer cottage, write down the answer in square centimeters.

A square is a positive quadrilateral (or rhombus) in which all angles are right and the sides are equal. Like any other regular polygon, square allowed to calculate perimeter and area. If area square already famous, then discover its sides, and after that and perimeter won't be difficult.

Instruction

1. Square square is found by the formula: S = a? This means that in order to calculate the area square, it is necessary to multiply the lengths of its 2 sides by each other. As a result, if you know the area square, then when extracting the root from this value, it is possible to find out the length of the side square.Example: area square 36 cm ?, in order to find out the side of this square, you need to extract Square root from the area value. So the side length of a given square 6 cm

2. For finding perimeter a square you need to add the lengths of all its sides. With the help of a formula, this can be expressed as follows: P \u003d a + a + a + a. If we extract the root from the area value square, and after that add the resulting value 4 times, then it is possible to find perimeter square .

3. Example: Given a square with an area of ​​49 cm². It needs to be discovered perimeter.Solution: First you need to take the root of the area square: ?49 = 7 cm Then, by calculating the length of the side square, it is allowed to calculate and perimeter: 7+7+7+7 = 28 cm Answer: perimeter square area 49 cm? is 28 cm

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

You will need

• Calculator

Instruction

1. If the square area is known, then in order to find the side of the square, you need to extract the square root from 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. Unit the side of a square will be the linear unit of length corresponding 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 obtained primitively in centimeters. Example: The area of ​​​​a square is 9 square meters. Find the length of the side of the square. Solution: a =?

2. In the case when the perimeter of the square is known, to determine the length of the side, it is necessary numerical value divide the perimeter by four (because the square has four sides of identical length): a = P / 4, where: a is the length of the side of the square; P is the perimeter of the square. The unit of measure for the side of the square will be the same linear unit of length as perimeter. Say, 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 the side of the square. Solution: a=20/4=5 Answer: The length of the side of the square is 5 meters.

3. If the length of the diagonal of a square is known, then the length of its side will be equal to the length of its diagonal divided by the square root of 2 (according to the Pythagorean theorem, because the adjacent sides of the square and the diagonal make up a rectangular isosceles triangle):a=d/?2(since 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 same as 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 \u003d 10 /? 10/?2, or approximately 1.071 meters.

The square is a beautiful and simple flat geometric figure. It is a rectangle with equal sides. How to discover perimeter square if the length of its side is known?

Instruction

1. Before everyone, it is worth remembering that perimeter is nothing more than the sum of the lengths of the sides of a geometric figure. The square we are considering has four sides. Moreover, by definition square, all these sides are equal to each other. From these premises it follows simple formula to find perimeter a square – perimeter square equal to the length of the side square multiplied by four: P = 4a, where a is the length of the side square .

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The perimeter is called the universal length the boundaries of the figure are more often than each on the plane. A square is a positive quadrilateral, either a rhombus, in which all angles are right, or a parallelogram, in which all sides and angles are equal.

You will need

• Geometry knowledge.

Instruction

1. Perimeter square is equal to the sum the length of its sides. Because a square, in its essence, is a quadrilateral, then it has four sides, which means that the perimeter is equal to the sum of the lengths of the four sides, or P = a + b + c + d.

2. A square, as can be seen from the definition, is a true geometric figure, which means that all its sides are equal. So a=b=c=d. Hence P = a+a+a+a or P = 4*a.

3. let side square is 4, that is, a=3. Then the perimeter or length square, according to the obtained formula, will be equal to P = 4*3 or P=12. The number 12 will be the length or, which is the same, the perimeter square .

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Note!
The perimeter of a square is invariably correct, as is any other length.

Useful advice
Similarly, it is possible to find the perimeter of a rhombus, because the square is a special case of a rhombus with right angles.

The perimeter characterizes the length of a closed silhouette. Like the area, it can be detected by other quantities given in the condition of the problem. Problems on finding the perimeter are extremely common in the school mathematics course.

Instruction

1. Knowing the perimeter and side of the figure, it is possible to find its other side, as well as the area. The perimeter itself, in turn, can be detected by several given sides or by the angle and sides, depending on the conditions of the problem. Also in some cases it is expressed through the area. The perimeter of a rectangle is especially primitive. Draw a rectangle with one side equal to a and a diagonal equal to d. Knowing these two values, use the Pythagorean theorem to find its other side, which is the width of the rectangle. Having found the width of the rectangle, calculate its perimeter in the following way: p=2(a+b). This formula is objective for all rectangles, since each of them has four sides.

2. Pay attention to the fact that the perimeter of a triangle in most problems is found if there is information about one of its corners. However, there are also problems in which all sides of the triangle are known, and then the perimeter can be calculated by simple summation, without the use of trigonometric calculations: p=a+b+c, where a, b and c are the sides. But such problems are rarely found in textbooks, because the method of solving them is clear. More difficult tasks of finding the perimeter of a triangle, solve in stages. Let's say draw an isosceles triangle, in which the base and the angle at it are famous. In order to find its perimeter, first find the sides a and b in the following way: b=c/2cos?. From the fact that a=b (isosceles triangle), make a further summary: a=b=c/2cos?.

3. Calculate the perimeter of a polygon in the same way, adding the lengths of all its sides: p=a+b+c+d+e+f and so on. If the polygon is positive and inscribed in or circumscribed by a circle, calculate the length of one of its sides, and then multiply by their number. Let's say, in order to find the sides of a hexagon inscribed in a circle, proceed as follows: a=R, where a is the side of the hexagon, equal to the radius of the circumscribed circle. Accordingly, if the hexagon is true, then its perimeter is equal to: p=6a=6R. If the circle is inscribed in a hexagon, then the side of the latter is: a=2r?3/3. Accordingly, find the perimeter of such a figure in the following way: p=12r?3/3.

Although the word "perimeter" comes from the Greek designation for a circle, it is customary to call it the total length of the boundaries of any flat geometric figure, including a square. The calculation of this parameter, as usual, is not difficult and can be carried out by several methods, depending on the famous initial data.

Instruction

1. If you know the length of the side of the square (t), then to find its perimeter (p) primitively increase this value four times: p=4*t.

2. If the length of the side is unknown, but the length of the diagonal (c) is given in the conditions of the problem, then this is enough to calculate the length of the sides, and, consequently, the perimeter (p) of the polygon. Use the Pythagorean theorem which states that the square of the length of the long side right triangle(hypotenuse) is equal to the sum of the squares of the lengths of the short sides (legs). In a right triangle made up of 2 adjacent sides of a square and connecting them extreme points segment, the hypotenuse coincides with the diagonal of the quadrilateral. From this it follows that the length of the side of the square is equal to the ratio of the length of the diagonal to the square root of two. Use this expression in the formula for calculating the perimeter from the previous step: p=4*c/?2.

3. If only the area (S) of a section of the plane bounded by the perimeter of the square is given, then this will be enough to determine the length of one side. Because the area of ​​any rectangle is equal to the product of the lengths of its adjacent sides, then to find the perimeter (p) take the square root of the area, and quadruple the total: p=4*?S.

4. If the radius of the circle described near the square (R) is known, then to find the perimeter of the polygon (p), multiply it by eight and divide the result by the square root of two: p=8*R/?2.

5. If the circle, the radius of which is kept, is inscribed in a square, then calculate its perimeter (p) by simply multiplying the radius (r) by eight: P=8*r.

6. If the square under consideration in the conditions of the problem is described by the coordinates of its vertices, then to calculate the perimeter you will need data on only 2 vertices belonging to one of the sides of the figure. Determine the length of this side, based on the same Pythagorean theorem for a triangle made up of itself and its projections on the coordinate axes, and quadruple the resulting result. Because the lengths of the projections on the coordinate axes are equal to the modulus of the differences between the corresponding coordinates of 2 points (X?; Y? and X?; Y?), then the formula can be written as follows: p=4*? ((X?-X?)? +(Y?-Y?)?).

In the general case, the perimeter is the length of the line that bounds the closed figure. For polygons, the perimeter is the sum of all side lengths. This value can be measured, and for many figures it is easy to calculate if the lengths of the corresponding elements are known.

You will need

• - ruler or tape measure;
• - strong thread;
• - roller rangefinder.

Instruction

1. In order to measure the perimeter of an arbitrary polygon, measure all its sides with a ruler or other measuring device, and then find their sum. Given a quadrilateral with sides of 5, 3, 7 and 4 cm, which are measured with a ruler, find the perimeter by adding them together P = 5 + 3 + 7 + 4 = 19 cm.

2. If the figure is arbitrary and includes not only straight lines, then measure its perimeter with a traditional rope or thread. To do this, position it so that it correctly repeats all the lines that bound the figure, and make a mark on it, if allowed, cut it primitively in order to avoid confusion. After that, using a tape measure or ruler, measure the length of the thread, it will be equal to the perimeter of this figure. Be sure to ensure that the thread repeats the line as accurately as possible for greater accuracy of the result.

3. Measure the perimeter of a difficult geometric figure with a roller rangefinder (curvimeter). To do this, a point is marked on the line, at which the rangefinder roller is installed and rolled along it, until it returns to the starting point. The distance measured by the roller rangefinder will be equal to the perimeter of the figure.

4. Calculate the perimeter of some geometric shapes. Say, in order to find the perimeter of any positive polygon ( convex polygon, whose sides are equal), multiply the length of the side by the number of angles or sides (they are equal). To find the perimeter right triangle with a side of 4 cm, multiply this number by 3 (P \u003d 4? 3 \u003d 12 cm).

5. To find the perimeter of an arbitrary triangle, add the lengths of all its sides. If all sides are not given, but there are angles between them, find them using the sine or cosine theorem. If two sides of a right triangle are famous, find the third side using the Pythagorean theorem and find their sum. Say, if it is known that the legs of a right triangle are 3 and 4 cm, then the hypotenuse will be equal to? (3? + 4?) = 5 cm. Then the perimeter P = 3 + 4 + 5 = 12 cm.

6. To find the perimeter of a circle, find the circumference of the circle that bounds it. To do this, multiply its radius r by the number??3.14 and the number 2 (P=L=2???r). If the diameter is known, consider that it is equal to two radii.

Perimeter polygon call a closed broken line made up of all its sides. Finding the length of this parameter is reduced to summing the lengths of the sides. If all the segments that form the perimeter of such a two-dimensional geometric figure have identical dimensions, the polygon is called true. In this case, the calculation of the perimeter is much simpler.

Instruction

1. In the simplest case, when we know the length of the side (a) of the correct polygon and the number of vertices (n) in it, to calculate the length of the perimeter (P), simply multiply these two values: P = a * n. Let's say the perimeter length of a true hexagon with a side of 15 cm should be equal to 15 * 6 = 90 cm.

2. Calculate the perimeter of this polygon along the known radius (R) of the circumscribed circle around it is also permissible. To do this, you will first have to express the length of the side using the radius and the number of vertices (n), and then multiply the resulting value by the number of sides. To calculate the length of a side, multiply the radius by the sine of pi divided by the number of vertices, and double the total: R*sin(?/n)*2. If you are more comfortable calculating the trigonometric function in degrees, replace Pi with 180°: R*sin(180°/n)*2. Calculate the perimeter by multiplying the obtained value by the number of vertices: Р = R*sin(?/n)*2*n = R*sin(180°/n)*2*n. Let's say if a hexagon is inscribed in a circle with a radius of 50 cm, its perimeter will have a length of 50*sin(180°/6)*2*6 = 50*0.5*12 = 300 cm.

3. By a similar method, it is possible to calculate the perimeter without knowing the length of the side of the positive polygon, if it is circumscribed about a circle with the famous radius (r). In this case, the formula for calculating the size of the side of the figure will differ from the previous one only involved trigonometric function. Replace the sine with the tangent in the formula to get the following expression: r*tg(?/n)*2. Or for calculations in degrees: r*tg(180°/n)*2. To calculate the perimeter, increase the resulting value by the number of times, equal to the number peaks polygon: P \u003d r * tg (? / n) * 2 * n \u003d r * tg (180 ° / n) * 2 * n. Let's say the perimeter of an octagon circumscribed near a circle with a radius of 40 cm will be approximately equal to 40*tg(180°/8)*2*8 ? 40 * 0.414 * 16 \u003d 264.96 cm.

A square is a geometric figure consisting of four sides of identical length and four right angles, each of which is equal to 90 °. Determining the area either perimeter a quadrangle, and any one, is required not only when solving problems in geometry, but also in Everyday life. This knowledge can become useful, say, during repairs when calculating the required number of materials - floor, wall or ceiling coverings, as well as for laying out lawns and beds, etc.

Instruction

1. To find the area of ​​a square, multiply the length by the width. Because in a square the length and width are identical, then the value of one side is quite square. Thus, the area of ​​a square is equal to the length of its squared side. The area unit can be square millimeters, centimeters, decimeters, meters, kilometers. To determine the area of ​​a square, you can use the formula S = aa, where S is square area, a- side of a square.

2. Example No. 1. The room has the shape of a square. How much laminate flooring (in sq.m.) will be needed in order to completely cover the floor if the length of one side of the room is 5 meters. Write down the formula: S \u003d aa. Substitute the data specified in the condition into it. Because a \u003d 5 m, therefore, the area will be equal to S (rooms) \u003d 5x5 \u003d 25 sq.m, which means S (laminate) \u003d 25 sq.m.

3. The perimeter is the total length of the figure's border. In a square, the perimeter is the length of all four, and identical, sides. That is, the perimeter of a square is the sum of all its four sides. To calculate the perimeter of a square, it is enough to know the length of one of its sides. The perimeter is measured in millimeters, centimeters, decimeters, meters, kilometers. To determine the perimeter, there is a formula: P \u003d a + a + a + a or P \u003d 4a, where P is the perimeter, and is the length of the side.

4. Example No. 2. For finishing work in a square-shaped room, ceiling plinths are required. Calculate the total length (perimeter) of the skirting boards if one side of the room is 6 meters. Write down the formula P \u003d 4a. Substitute the data indicated in the condition into it: P (rooms) \u003d 4 x 6 \u003d 24 meters. Consequently, the length of the ceiling plinths will also be 24 meters.

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Note!
The following definitions are objective for a square: A square is a rectangle, one that has sides equal to each other. A square is a special kind of rhombus, in which all of the angles are 90 degrees. Being a positive quadrilateral, it is possible to describe or inscribe a circle around the square. The radius of a circle inscribed in a square can be found by the formula: R = t / 2, where t is the side of the square. If the circle is described around it, then its radius is found as follows: R = (? 2 * t) / 2 Based on these formulas, it is allowed derive new ones to find the perimeter of the square: P = 8*R, where R is the radius of the inscribed circle; P = 4*?2*R, where R is the radius of the circumscribed circle. A square is a unique geometric figure, because it is unconditionally symmetrical, independently on how and where to draw the axis of symmetry.

Many people remember what a square is from school course. This quadrilateral, which is regular, has absolutely equal angles and sides. Looking around, you can see that we are surrounded by many squares. Every day we encounter them, and sometimes it becomes necessary to find the area and perimeter of this geometric figure. Calculating these values ​​is easy if you take a few minutes to watch this video tutorial explaining simple rules carrying out calculations.

Tutorial video “How to find the area and perimeter of a square”

What you need to know about the square?

Before proceeding with the calculations, you need to know some important information about this figure, including:

• all sides of a square are equal;
• all corners of the square are right;
• the area of ​​a square is a way of calculating how much space a figure takes up in two-dimensional space;
• two-dimensional space is a sheet of paper or a computer screen where a square is drawn;
• the perimeter is not an indicator of the fullness of the figure, but allows you to work with its sides;
• perimeter is the sum of all sides of a square;
• when calculating the perimeter, we operate in one-dimensional space, which means fixing the result in meters, not square meters (area).

How to find the area of ​​a square?

The calculation of the area of ​​​​a given figure can be simply and easily explained with an example:

• suppose that the side of the square is 8 meters;
• to calculate the area of ​​any rectangle, you need to multiply the value of one of its sides by the other (8 x 8 \u003d 64);
• since we multiply meters by meters, the result is square meters(m2).

How to find the perimeter of a square?

Knowing that all sides of a given rectangle are equal, you need to do the following manipulations to calculate its perimeter:

• add up all four sides of the square (8 + 8 + 8 + 8 = 32);
• the resulting value will be the perimeter of the square, fixed in meters.

All formulas and calculations given in this article are applicable to any rectangle. It is important to remember that when it comes to other rectangles that are not correct, the value of the sides will be different, for example 4 and 8 meters. This means that in order to find the area of ​​such a rectangle, it will be necessary to multiply the sides of the figure that are different in value, and not the same.

It must also be remembered that the area is measured in square meters, and the perimeter in simple meters. If the perimeter is drawn as one long line, then its value will not change, which indicates that the calculations are carried out in one-dimensional space.

Area is measured in two-dimensional space, as indicated by square meters, which we get by multiplying meters by meters. The area is an indicator of the fullness of a geometric figure, and tells us how much imaginary coverage is needed in order to fill a square or other rectangle.

Simple explanations of the video lesson will allow you to quickly calculate the area and perimeter of not only a square, but also any rectangle. This knowledge of the school course will be useful during the repair of the house or in the garden.

Calculating the perimeter of a square is an important skill. And it's not just about schoolwork. After all, with the help of simple mathematical operations, you can easily calculate the amount of building material you need. For example, to install a fence around the perimeter of a square area or wallpapering in a square room.

To find the perimeter of a square, you need to know the value of one of the sides, the area or radius of the circumscribed circle. Let's consider these methods in more detail.

How to find the perimeter of a square given one side of the square

• The perimeter of a figure is the sum of all its sides. Since a square has only 4 sides, its perimeter is:
  P \u003d a + b + c + d,
  where P is the perimeter,
  a, c, c, e - sides.
• Knowing that all sides of a square are equal, we simplify the formula:
  P = 4a,
  where a is one of the sides,
  4 is the sum of the sides.
• Example solution: if the side is 7, then
  P \u003d 4 * 7 \u003d 28.

How to find the perimeter of a square given the area of ​​a square

• The area of ​​a square is calculated by the formula:
  S \u003d a * a \u003d a²,
  where S is the area,
  a - any side.
• Let's rewrite the formula:
  a² = S,
  a = √S.
  Example solution: if the area is 121, then
  a = √121 = 11.
• Knowing the side of the square, we can find the perimeter:
  P = 4*a.
• Solution example: P \u003d 4 * 11 \u003d 44.

How to find the perimeter of a square given the radius of the circumscribed circle

Suppose we are given a square and know the radius of a circle that describes it from all sides. If we draw a diagonal between the opposite corners of the square, then we get 2 triangles with right angles. In this case, it is a sin not to use the Pythagorean theorem, which says: "The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse."

What else do we know:

• The sides in and with in 2 triangles are equal, since these are the sides of the square. They are also skates.
• Triangles have a common hypotenuse a, which is also the diameter of the circle.
• The diameter is equal to two radii (2r).

Let's start finding the perimeter:

• According to the Pythagorean theorem:
  b² + c² = a²,
  where in and c are the legs of a right triangle,
  a is the hypotenuse.
• Knowing that a (hypotenuse) \u003d 2r, and b \u003d c, we simplify the formula:
  in² + in² = (2r)²,
  2в² = 4(r)², reduce by 2:
  в² = 2(r)²,
  c = √2r, where
  c is the side of the square.
• Since the perimeter of a square is equal to the sum of the sides, we modify the formula:
  Р = 4√2r,
  where P is the desired perimeter,
  4 - the sum of the sides,
  √2r - side length.
• Let's simplify the formula:
  P = 4√2 * 4√r,
  P = 5.657r,
  where P is the desired perimeter,
  r is the radius of the circle.

Solution example:

If the radius of the circle is 20:

P \u003d 5.657 * 20 \u003d 113.14.

The numbers are quickly forgotten, but the problem can always be solved using the Pythagorean theorem:

in² + in² \u003d (2 * 20)²,
2v² = 40²,
2v² \u003d 1600, divided by 2:
in² = 800,
c = √800,
c = 28.28,
where s is one side.
So,
P \u003d 4 * 28.29,
P = 113.14.

There are many ways to find the perimeter of a square, but they all come down to the fact that the perimeter is equal to the sum of all sides.