Perimeter, area and volume. How to find the perimeter of a square if its area is known What you need to know about a square

The relationship between the radius of a circle and the side length of a square. The distance from the center of the circumscribed circle to the vertex of the square inscribed in it is equal to the radius of the circle. To find the side of a square s, you need to divide the square diagonally into 2 right triangles. Each of these triangles will have equal sides a And b and common hypotenuse With, equal to twice the radius of the circumscribed circle ( 2r).

Use the Pythagorean theorem to find the side of a square. The Pythagorean theorem states that in any right triangle with legs A And b and hypotenuse With: a 2 + b 2 = c 2. Since in our case A = b(remember we are looking at a square!) and we know that c = 2r, then we can rewrite and simplify this equation:

• a 2 + a 2 = (2r) 2 ""; Now let's simplify this equation:
• 2a 2 = 4(r) 2; Now let's divide both sides of the equation by 2:
• (a 2) = 2(r) 2; now let's extract Square root from both sides of the equation:
• a = √(2r). Thus s = √ (2r).

1. Multiply the found side of the square by 4 to find its perimeter. In this case, the perimeter of the square is: P = 4√(2r). This formula can be rewritten as follows: Р = 4√2 * 4√r = 5.657r, where r is the radius of the circumscribed circle.

2. Example. Consider a square inscribed in a circle of radius 10. This means that the diagonal of the square is 2 * 10 = 20. Using the Pythagorean theorem, we get: 2(a 2) = 20 2, that is 2a 2 = 400. Now we divide both sides of the equation by 2 and get: a 2 = 200. Now let's take the square root of both sides of the equation and get: a = 14.142. Multiply this value by 4 and calculate the perimeter of the square: P=56.57.

   • Note that you could get the same result by simply multiplying radius(10) by 5.657: 10 * 5,567 = 56,57 ; but this method is difficult to remember, so it is better to use the calculation process described above.

Many people remember what a square is from school. 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 come across them, and sometimes the need arises 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.

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

What do you need to know about the square?

Before you begin making calculations, you need to know some important information about this figure, including:

• all sides of the square are equal;
• all corners of a square are right;
• The area of ​​a square is a way of calculating how much space a shape 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 the square;
• When calculating the perimeter, we operate with one-dimensional space, which means recording the result in meters, not square meters (area).

How to find the area of ​​a square?

Calculating the area of ​​a given figure can be simply and easily explained using an example:

• Let's assume that the side of the square is 8 meters;
• to calculate the area of ​​any rectangle, you need to multiply the value of one side by the other (8 x 8 = 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, recorded in meters.

All formulas and calculations given in this article are applicable for any rectangle. It is important to remember that when it comes to other rectangles that are not regular, the sides will have different values, for example 4 and 8 meters. This means that 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 ones.

It is also necessary to remember 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 dimensions, as indicated by square meters, which we get by multiplying meters by meters. Area is an indicator of the fullness of a geometric figure, and tells us how much imaginary coverage is needed 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 from the school course will be useful when renovating a house or garden.

This material contains geometric shapes with measurements. The measurements given are approximate and may not correspond to actual measurements. Lesson content

Perimeter of a geometric figure

The perimeter of a geometric figure is the sum of all its sides. To calculate the perimeter, you need to measure each side and add the measurements.

Let's calculate the perimeter of the following figure:

This is a rectangle. We will talk about this figure in more detail later. Now let's just calculate the perimeter of this rectangle. Its length is 9 cm and width 4 cm.

A rectangle has opposite sides that are equal. This can be seen in the figure. If the length is 9 cm and the width is 4 cm, then the opposite sides will be 9 cm and 4 cm respectively:

Let's find the perimeter. To do this, let's add all the sides. You can add them in any order, since rearranging the places of the terms does not change the sum. The perimeter is often indicated by a capital Latin letter P(English) perimeters). Then we get:

P= 9 cm + 4 cm + 9 cm + 4 cm = 26 cm.

Since the opposite sides of a rectangle are equal, finding the perimeter is written shorter - add the length and width, and multiply it by 2, which will mean “repeat length and width twice”

P= 2 × (9 + 4) = 18 + 8 = 26 cm.

A square is the same as a rectangle, but with all sides equal. For example, let’s find the perimeter of a square with a side of 5 cm. The phrase "with the side 5cm" need to understand how "The length of each side of the square is equal to 5cm"

To calculate the perimeter, add up all the sides:

P= 5 cm + 5 cm + 5 cm + 5 cm = 20 cm

But since all sides are equal, the perimeter calculation can be written as a product. The side of the square is 5 cm, and there are 4 such sides. Then this side, equal to 5 cm, must be repeated 4 times

P= 5 cm × 4 = 20 cm

Area of ​​a geometric figure

The area of ​​a geometric figure is a number that characterizes the size of this figure.

It should be clarified that in this case we are talking about area on a plane. In geometry, a plane is any flat surface, for example: a sheet of paper, a plot of land, a table surface.

Area is measured in square units. Square units mean squares whose sides are equal to one. For example, 1 square centimeter, 1 square meter or 1 square kilometer.

To measure the area of ​​a figure means to find out how many square units are contained in this figure.

For example, the area of ​​the following rectangle is three square centimeters:

This is because this rectangle contains three squares, each of which has a side equal to one centimeter:

On the right is a square with a side of 1 cm (in this case it is a square unit). If we look at how many times this square fits into the rectangle shown on the left, we will find that it fits into it three times.

The following rectangle has an area equal to six square centimeters:

This is because this rectangle contains six squares, each of which has a side equal to one centimeter:

Let's say you needed to measure the area of ​​the following room:

Let's decide in which squares we will measure the area. In this case, it is convenient to measure the area in square meters:

So, our task is to determine how many such squares with a side of 1 m are contained in the original room. Let's fill the entire room with this square:

We see that a square meter is contained in a room 12 times. This means that the area of ​​the room is 12 square meters.

Area of ​​a rectangle

In the previous example, we calculated the area of ​​the room by sequentially checking how many times it contains a square whose side is equal to one meter. The area was 12 square meters.

The room was a rectangle. The area of ​​a rectangle can be calculated by multiplying its length and width.

To calculate the area of ​​a rectangle, you need to multiply its length and width.

Let's return to the previous example. Let's say we measured the length of the room with a tape measure and it turned out that the length was 4 meters:

Now let's measure the width. Let it be 3 meters:

Multiply the length (4 m) by the width (3 m).

4 × 3 = 12

Like last time, we get twelve square meters. This is explained by the fact that by measuring the length, we thereby find out how many times a square with a side equal to one meter can be placed into this length. Let's fit four squares into this length:

Then we determine how many times this length can be repeated with stacked squares. We find out this by measuring the width of the rectangle:

Square area

A square is the same as a rectangle, but with all sides equal. For example, the following figure shows a square with a side of 3 cm. The phrase "a square with a side 3cm" means all sides are 3 cm

The area of ​​a square is calculated in the same way as the area of ​​a rectangle - the length is multiplied by the width.

Calculate the area of ​​a square with a side of 3 cm. Multiply the length 3 cm by the width 3 cm

In this case, it was necessary to find out how many squares with a side of 1 cm are contained in the original square. The original square contains nine squares with a side of 1 cm. Indeed, this is so. A square with a side of 1 cm enters the original square nine times:

By multiplying the length by the width, we got the expression 3 × 3, and this is the product of two identical factors, each of which is equal to 3. In other words, the expression 3 × 3 represents the second power of the number 3. This means that the process of calculating the area of ​​a square can be written as a power 3 2.

Therefore, the second power of the number is called square the number. When calculating the second power of a number a, a person thereby finds the area of ​​a square with side a. The operation of raising a number to the second power is also called squaring.

Designations

The area is indicated by a capital Latin letter S(English) Square- square). Then the area of ​​a square with side a cm will be calculated according to the following rule

S = a 2

Where a- length of the side of the square. The second degree indicates that two identical factors are multiplied, namely length and width. It was said earlier that all sides of a square are equal, which means the length and width of the square are equal, expressed through the letter a .

If the task is to determine how many squares with a side of 1 cm are contained in the original square, then cm 2 should be specified as the units of area. This designation replaces the phrase "square centimeter" .

For example, let's calculate the area of ​​a square with a side of 2 cm.

This means that a square with a side of 2 cm has an area equal to four square centimeters:

If the task is to determine how many squares with a side of 1 m are contained in the original square, then m 2 should be specified as the units of measurement. This designation replaces the phrase "square meter" .

Calculate the area of ​​a square with a side of 3 meters

This means that a square with a side of 3 m has an area equal to nine square meters:

Similar notation is used when calculating the area of ​​a rectangle. But the length and width of the rectangle can be different, so they are denoted by different letters, for example a And b. Then the area of ​​the rectangle, length a and width b is calculated according to the following rule:

S = a × b

As in the case of a square, the units of measurement for the area of ​​a rectangle can be cm 2, m 2, km 2. These designations replace phrases "square centimeter", "square meter", "square kilometer" respectively.

For example, let's calculate the area of ​​a rectangle with a length of 6 cm and a width of 3 cm

This means that a rectangle 6 cm long and 3 cm wide has an area equal to eighteen square centimeters:

It is allowed to use the phrase as a unit of measurement "square units" . For example, record S = 3 sq. units means that the area of ​​a square or rectangle is equal to three squares, each of which has a unit side (1 cm, 1 m or 1 km).

Conversion of area units

Area units can be converted from one unit of measurement to another. Let's look at a few examples:

Example 1. Express 1 square meter in square centimeters.

1 square meter is a square with a side of 1 m. That is, all four sides have a length equal to one meter.

But 1 m = 100 cm. Then all four sides also have a length equal to 100 cm

Let's calculate new square this square. Multiply the length of 100 cm by the width of 100 cm or square the number 100

S = 100 2 = 10,000 cm 2

It turns out that there are ten thousand square centimeters per square meter.

1 m2 = 10,000 cm2

This allows you to multiply any number of square meters by 10,000 in the future and get the area expressed in square centimeters.

To convert square meters to square centimeters, you need to multiply the number of square meters by 10,000.

To convert square centimeters to square meters, you need to, on the contrary, divide the number of square centimeters by 10,000.

For example, let's convert 100,000 cm 2 to square meters. In this case, you can reason like this: “ If 10,000 cm 2 this is one square meter, then how many times 100,000 cm 2 will contain 10,000 cm 2 "

100,000 cm 2: 10,000 cm 2 = 10 m 2

Other units of measurement can be converted in the same way. For example, let's convert 2 km 2 to square meters.

One square kilometer is a square with a side of 1 km. That is, all four sides have a length equal to one kilometer. But 1 km = 1000 m. This means that all four sides of the square are also equal to 1000 m. Let's find the new area of ​​the square, expressed in square meters. To do this, multiply the length of 1000 m by the width of 1000 m or square the number 1000

S = 1000 2 = 1,000,000 m2

It turns out that there are one million square meters per square kilometer:

1 km 2 = 1,000,000 m 2

This makes it possible in the future to multiply any number of square kilometers by 1,000,000 and obtain the area expressed in square meters.

To convert square kilometers to square meters, you need to multiply the number of square kilometers by 1,000,000.

So, let's return to our task. It was necessary to convert 2 km 2 into square meters. Multiply 2 km 2 by 1,000,000

2 km 2 × 1,000,000 = 2,000,000 m2

And to convert square meters to square kilometers, you need to, on the contrary, divide the number of square meters by 1,000,000.

For example, let's convert 3,500,000 m2 to square kilometers. In this case, you can reason like this: “ If 1,000,000 m2 this is one square kilometer, then how many times 3,500,000 m2 will contain 1,000,000 m2"

3,500,000 m2: 1,000,000 m2 = 3.5 km2

Example 2. Express 7 m2 in square centimeters.

Multiply 7 m2 by 10,000

7 m 2 = 7 m 2 × 10,000 = 70,000 cm 2

Example 3. Express 5 m 2 13 cm 2 in square centimeters.

5 m 2 13 cm 2 = 5 m 2 × 10,000 + 13 cm 2 = 50,013 cm 2

Example 4. Express 550,000 cm 2 in square meters.

Let's find out how many times 550,000 cm 2 contains 10,000 cm 2. To do this, divide 550,000 cm 2 by 10,000 cm 2

550,000 cm 2: 10,000 cm 2 = 55 m 2

Example 5. Express 7 km 2 in square meters.

Multiply 7 km 2 by 1,000,000

7 km 2 × 1,000,000 = 7,000,000 m2

Example 6. Express 8,500,000 m2 in square kilometers.

Let's find out how many times 8,500,000 m2 contains 1,000,000 m2. To do this, divide 8,500,000 m2 by 1,000,000 m2

8,500,000 m2 × 1,000,000 m2 = 8.5 km2

Units of land area measurement

It is convenient to measure the area of ​​small plots of land in square meters.

The areas of larger land plots are measured in ares and hectares.

Ar(abbreviated: a) is an area equal to one hundred square meters (100 m2). Due to the frequent distribution of such an area (100 m2), it began to be used as a separate unit of measurement.

For example, if it is said that the area of ​​a field is 3 a, then you need to understand that these are three squares with an area of ​​100 m2 each, that is:

3 a = 100 m 2 × 3 = 300 m 2

among the people ar often call hundred, since ap is equal to a square with an area of ​​100 m 2. Examples:

1 hundred square meters = 100 m 2

2 acres = 200 m 2

10 acres = 1000 m2

Hectare(abbreviated: ha) is an area equal to 10,000 m 2. For example, if it is said that the area of ​​a forest is 20 hectares, then you need to understand that these are twenty squares with an area of ​​10,000 m2 each, that is:

20 ha = 10,000 m 2 × 20 = 200,000 m 2

Rectangular parallelepiped and cube

A rectangular parallelepiped is a geometric figure consisting of faces, edges and vertices. The figure shows a rectangular parallelepiped:

Shown in yellow edges parallelepiped, black - ribs, red - peaks.

A rectangular parallelepiped has length, width and height. The figure shows where the length, width and height are:

A parallelepiped whose length, width and height are equal is called. The figure shows a cube:

Volume of a geometric figure

Volume of a geometric figure is a number that characterizes the capacity of a given figure.

Volume is measured in cubic units. Cubic units mean cubes with a length of 1, a width of 1 and a height of 1. For example, 1 cubic centimeter or 1 cubic meter.

To measure the volume of a figure means to find out how many cubic units fit into this figure.

For example, the volume of the following rectangular parallelepiped is twelve cubic centimeters:

This is because this parallelepiped fits twelve cubes 1 cm long, 1 cm wide and 1 cm high:

Volume is indicated by a capital Latin letter V. One of the units for measuring volume is cubic centimeter (cm3). Then the volume V the parallelepiped we considered is 12 cm 3

V= 12 cm 3

The volume of any parallelepiped is calculated as follows: multiply its length, width and height.

The volume of a rectangular parallelepiped is equal to the product of its length, width and height.

V=abc

Where, a- length, b- width, c- height

So, in the previous example, we visually determined that the volume of the parallelepiped is 12 cm 3. But you can measure the length, width and height of a given parallelepiped and multiply the measurement results. We will get the same result

Volume is calculated in the same way as volume rectangular parallelepiped- multiply length, width and height.

For example, let's calculate the volume of a cube whose length is 3 cm. The length, width and height of a cube are equal. If the length is 3 cm, then the width and height of the cube are equal to the same three centimeters:

We multiply the length, width, height and get a volume equal to twenty-seven cubic centimeters:

V= 3 × 3 × 3 = 27 cm³

Indeed, the original cube contains 27 cubes 1 cm long

When calculating the volume of a given cube, we multiplied the length, width and height. The result is the product 3 × 3 × 3. This is the product of three factors, each of which is equal to 3. In other words, the product 3 × 3 × 3 is the third power of the number 3 and can be written as 3 3.

V= 3 3 = 27 cm 3

Therefore, the third power of a number is called cubed numbers. When calculating the third power of a number a, a person thereby finds the volume of a cube, length a. The operation of raising a number to the third power is also called cubed.

Thus, the volume of a cube is calculated according to the following rule:

V=a 3

Where a— cube length.

Cubic decimeter. Cubic meter

Not all objects in our world are conveniently measured in cubic centimeters. For example, it is more convenient to measure the volume of a room or house in cubic meters (m3). And it is more convenient to measure the volume of a tank, aquarium or refrigerator in cubic decimeters (dm 3).

Another name for one cubic decimeter is one liter.

1 dm 3 = 1 liter

Conversion of volume units

Volume units can be converted from one unit of measurement to another. Let's look at a few examples:

Example 1. Express 1 cubic meter in cubic centimeters.

One cubic meter is a cube with a side of 1 m. The length, width and height of this cube are equal to one meter.

But 1 m = 100 cm. This means that the length, width and height are also equal to 100 cm

Let's calculate the new volume of the cube, expressed in cubic centimeters. To do this, multiply its length, width and height. Or let's cube the number 100:

V = 100 3 = 1,000,000 cm 3

It turns out that there are one million cubic centimeters per cubic meter:

1 m 3 = 1,000,000 cm 3

This allows you to multiply any number of cubic meters by 1,000,000 in the future and get the volume expressed in cubic centimeters.

To convert cubic meters to cubic centimeters, you need to multiply the number of cubic meters by 1,000,000.

And to convert cubic centimeters to cubic meters, you need to, on the contrary, divide the number of cubic centimeters by 1,000,000.

For example, let's convert 300,000,000 cm 3 to cubic meters. In this case, you can reason like this: “ If 1,000,000 cm 3 this is one cubic meter, then how many times 300,000,000 cm3 will contain 1,000,000 cm 3 "

300,000,000 cm 3: 1,000,000 cm 3 = 300 m 3

Example 2. Express 3 m 3 in cubic centimeters.

Multiply 3 m 3 by 1,000,000

3 m 3 × 1,000,000 = 3,000,000 cm 3

Example 3. Express 60,000,000 cm 3 in cubic meters.

Let's find out how many times 60,000,000 cm 3 contains 1,000,000 cm 3. To do this, divide 60,000,000 cm 3 by 1,000,000 cm 3

60,000,000 cm 3: 1,000,000 cm 3 = 60 m 3

The capacity of a tank, can or canister is measured in liters. A liter is also a unit of volume. One liter is equal to one cubic decimeter.

1 liter = 1 dm 3

For example, if the capacity of a jar is 1 liter, this means that the volume of this jar is 1 dm 3. When solving some problems it may be useful skill Convert liters to cubic decimeters and vice versa. Let's look at a few examples.

Example 1. Convert 5 liters to cubic decimeters.

To convert 5 liters to cubic decimeters, just multiply 5 by 1

5 l × 1 = 5 dm 3

Example 2. Convert 6000 liters to cubic meters.

Six thousand liters is six thousand cubic decimeters:

6000 l × 1 = 6000 dm 3

Now let's convert these 6000 dm 3 into cubic meters.

The length, width and height of one cubic meter are equal to 10 dm

If we calculate the volume of this cube in decimeters, we get 1000 dm 3

V= 10 3 = 1000 dm 3

It turns out that one thousand cubic decimeters corresponds to one cubic meter. And to determine how many cubic meters correspond to six thousand ml cubic decimeters, you need to find out how many times 6,000 dm 3 contains 1,000 dm 3

6,000 dm 3 : 1,000 dm 3 = 6 m 3

This means 6000 l = 6 m3.

Table of squares

In life, you often have to find the area of ​​various squares. To do this, each time you need to raise the original number to the second power.

First 99 squares natural numbers have already been calculated and entered into a special table called table of squares.

The first row of this table (numbers from 0 to 9) is the original number, and the first column (numbers from 1 to 9) is the original number.

For example, let's find the square of the number 24 using this table. The number 24 is made up of the digits 2 and 4. More precisely, the number 24 is made up of two tens and four ones.

So, we select the number 2 in the first column of the table (the tens column), and select the number 4 in the first row (the units row). Then, moving to the right of number 2 and down from number 4, we will find the intersection point. As a result, we will find ourselves in the position where the number 576 is located. This means that the square of the number 24 is the number 576

24 2 = 576

Cube table

As with squares, the cubes of the first 99 natural numbers have already been calculated and entered into a table called table of cubes.

Calculate the volume of a rectangular parallelepiped whose length is 6 cm, width 4 cm, height 3 cm. Problem 7. Areas land plot sown with wheat and flax are proportional to the numbers 4 and 5. On what area is wheat sown if 15 hectares are sown under flax

Solution

The number 4 reflects the area sown with wheat. And the number 5 reflects the area sown with flax.
It is said that the areas sown with wheat and flax are proportional to these numbers.

Simply put, by how many times the numbers 4 or 5 change, by how many times the area sown with wheat or flax will change. 15 hectares are sown with flax. That is, the number 5, which reflects the area sown with flax, has changed 3 times.

Then the number 4, which reflects the area sown with wheat, needs to be increased three times

4 × 3 = 12 hectares

Answer: 12 hectares are sown with wheat.

Problem 8. The length of the granary is 42 m, the width is equal to the length, and the height is 0.1 times the length. Determine how many tons of grain the granary can hold if 1 m3 weighs 740 kg.

Solution

Let's determine how many liters per minute flow through the second pipe:

25 l/min × 0.75 = 18.75 l/min

Let's determine how many liters per minute flow into the pool through both pipes:

25 l/min + 18.75 l/min = 43.75 l/min

Let's determine how many liters of water will be poured into the pool in 13 hours 32 minutes

43.75 × 13 h 32 min = 43.75 × 812 min = 35,525 l

1 l = 1 dm 3

35,525 l = 35,525 dm 3

Let's convert cubic decimeters to cubic meters. This will allow you to calculate the volume of the pool:

35,525 dm 3: 1000 dm 3 = 35.525 m 3

Knowing the volume of the pool, you can calculate the height of the pool. Let's substitute it into the literal equation V=abc the values ​​we have. Then we get:

V = 35,525
a = 5.8
b = 3.5
c= x

35.525 = 5.8 × 3.5 × x
35.525 = 20.3 × x
x= 1.75 m

c = 1.75

Answer: The height (depth) of the pool is 1.75 m.

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

Instructions:

• First, let's determine that perimeter is 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 side length and diagonal

• Because the perimeter of a square is determined by the sum of the lengths of all its sides, and the sides of a given figure are equal, then the value of this value can be calculated 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- This perimeter of a square And A – side length.
• In addition, depending on the conditions of the problem, the perimeter of a square can be calculated by multiplying the length of its diagonal by two roots of two: P = 2√2 * d , Where R- This perimeter of a square And d- his diagonal.
• Some tasks require finding perimeter of a square knowing him square . This will also not be difficult to do. The area of ​​a given figure is equal to the length of its side squared: S = a 2 , Where S – area of ​​the square And A –length of its side. Or the area is equal to square value the length of its diagonal divided by two: S = d 2 /2 , Where S- still the same square And d – diagonal of a square.
• 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 a 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 = ½ a , Where R in – inscribed circle radius And A – side of a square.
• Circumcircle passes through all the vertices of the square and its radius is equal to half the length of the diagonal: R o = ½ d , Where R o – this radius of a circle circumscribed around a square And d- his diagonal.
• Therefore, in the first case, the perimeter will be calculated using the formula: Р = 8 R in , and in the second: P = 4 x √2 x R o .

Using websites and an online calculator

• If for some reason you suddenly forget the formulas, then the Internet will help you refresh your knowledge. Go to your browser, open the search engine page and enter the appropriate query in the window, for example: “ perimeter of a square formula" The system will display a huge number sites for reference purposes that will help you with this issue, and will also allow you to cope with solving problems related to other geometric shapes.
• In addition, if you do not want to understand formulas and calculate values ​​yourself, then you can use the services Internet calculators . An example would be a website. Chapter " Formulas for the perimeter of geometric figures"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 window below on the basis of which you are going to calculate perimeter of a square(side or diagonal), and then enter the available data. The system will issue result , guided by 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 also look for more convenient or educational help sites.
• If you cannot figure out the process of solving the problem, then here you can turn to people who are good at solving mathematical exercises for help. They can always be found on the corresponding forums , for example, or.

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 perimeter won't be difficult.

Instructions

1. Square square is found by the formula: S = a? This means that in order to calculate the area square, you need to multiply the lengths of its 2 sides by each other. As a consequence, if you know the area square, then when extracting the root from a given value, you can 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 take the square root of the area value. Thus, the length of the side of a given square 6 cm

2. To find perimeter A square you need to add up the lengths of all its sides. With the help of a formula, this can be expressed as follows: P = a+a+a+a. If you take the root of the area value square, and after that add the resulting value 4 times, then you can detect perimeter square .

3. Example: Given a square with an area of ​​49 cm?. Need to discover it perimeter.Solution: First you need to extract the root of the area square: ?49 = 7 cmThen, calculating the length of the side square, it is possible to calculate and perimeter: 7+7+7+7 = 28 cmAnswer: perimeter square area 49 cm? is 28 cm

Often in geometric problems it is necessary to find the length of the 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 known, then in order to find the side of the square, you need to take 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. Unit measuring the side of a square will be a linear unit of length, corresponding to a unit of area. Say, if the area of ​​a square is given in square centimeters, then the length of its side will be primitively in centimeters. Example: The area of ​​a square is 9 square meters. Find the length of the side of the 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 known, 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 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 measurement for the side of a square will be the same linear unit of length as the 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, 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 form a right 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 unit of measurement for the side of the square will be the same unit of length 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 = 10/?2, or approximately: 7.071 Answer: The length of the side of the square is 10/?2, or approximately 1.071 meters.

A square is a beautiful and simple flat geometric figure. This is a rectangle with equal sides. How to detect perimeter square, if the length of its side is known?

Instructions

1. Before everyone else, 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 follows a simple formula for finding perimeter A square – perimeter square equal to length sides 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 frequent than each one on the plane. A square is a positive quadrilateral or a rhombus in which all angles are right, or a parallelogram in which all sides and angles are equal.

You will need

• Knowledge of geometry.

Instructions

1. Perimeter square equal to the sum of the lengths of its sides. Because a square, in its essence, is a quadrilateral, it has four sides, which means 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 regular geometric figure, which means that all its sides are equal. So a=b=c=d. Consequently, P = a+a+a+a or P = 4*a.

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

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Note!
The perimeter of a square is invariably the correct value, like any other length.

Helpful advice
In a similar way, it is possible to determine the perimeter of a rhombus, because a square is a special case of a rhombus with right angles.

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

Instructions

1. Knowing the perimeter and side of a figure, you can discover its other side, as well as its area. The perimeter itself, in turn, can be detected along several specified sides or along an angle and sides, depending on the conditions of the problem. Also in some cases it is expressed through 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 quantities, 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 as follows: 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 in most problems the perimeter of a triangle is found only if there is information about only one of its angles. However, there are also problems in which all the 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 for solving them is clear. Solve more difficult problems of finding the perimeter of a triangle step by step. Let's say, draw an isosceles triangle whose base and angle are known. In order to find its perimeter, first find sides a and b as follows: b=c/2cos?. From the fact that a=b (isosceles triangle), make a further result: a=b=c/2cos?.

3. Calculate the perimeter of the polygon in a similar 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 a circle or described around it, 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 correct, then its perimeter is equal to: p=6a=6R. If a circle is inscribed in a hexagon, then the side of the latter is equal to: 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 refer to the total length of the boundaries of any flat geometric figure, including a square. Calculating this parameter, as usual, is not difficult and can be carried out using several methods, depending on the known initial data.

Instructions

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

2. If the length of the side is unknown, but in the conditions of the problem the length of the diagonal (c) is given, 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 longest 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. It follows from this that the length of the side of a square is equal to the ratio of the length of the diagonal to the square root of two. Use this expression in the formula to calculate the perimeter from the previous step: p=4*c/?2.

3. If only the area (S) of a section of the plane limited 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 is known (R), then to find the perimeter of the polygon (p), multiply it by eight and divide the resulting total by the square root of two: p=8*R/?2.

5. If the circle whose radius 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 in question in the problem conditions 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 composed of itself and its projections on the coordinate axes, and increase the resulting total by four times. Because the lengths of the projections onto 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 general, the perimeter is the length of the line that limits a closed figure. For polygons, the perimeter is the sum of all side lengths. This value can be measured, and for many figures it can be easily calculated if the lengths of the corresponding elements are known.

You will need

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

Instructions

1. To measure the perimeter of an arbitrary polygon, use a ruler or other measuring device to measure all its sides, and then find their sum. If 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 more than just straight lines, then measure its perimeter with a traditional rope or thread. To do this, position it so that it correctly follows all the lines limiting the figure, and make a mark on it; if possible, trim it primitively in order to avoid confusion. After this, 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 follows the line as accurately as possible for greater accuracy of the result.

3. Measure the perimeter of a difficult geometric figure with a roller range finder (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 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 (a convex polygon whose sides are equal), multiply the length of the side by the number of angles or sides (they are equal). In order to find the perimeter of a regular triangle with a side of 4 cm, multiply this number by 3 (P = 4? 3 = 12 cm).

5. To find the perimeter of an arbitrary triangle, add up 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 known, find the third using the Pythagorean theorem and find their sum. Let's say, if it is known that the legs of a right triangle are equal to 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 that limits 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 called closed broken line, composed of all its sides. Finding the length of this parameter comes down to summing the lengths of the sides. If all the segments forming the perimeter of such a two-dimensional geometric figure have identical dimensions, the polygon is called true. In this case, calculating the perimeter is much simpler.

Instructions

1. In the simplest case, when the length of 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 quantities: P = a*n. Let's say the perimeter length of a regular hexagon with a side of 15 cm should be equal to 15 * 6 = 90 cm.

2. Calculate the perimeter of such polygon along the known radius (R) of the circle described 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 side length, 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 resulting value by the number of vertices: P = R*sin(?/n)*2*n = R*sin(180°/n)*2*n. 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. Using a similar method it is possible to calculate the perimeter without knowing the length of the positive side polygon, if it is described around a circle with a 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 = r*tg(?/n)*2*n = r*tg(180°/n)*2*n. Let's say, the perimeter of an octagon described near a circle with a radius of 40 cm will be approximately equal to 40*tg(180°/8)*2*8? 40*0.414*16 = 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°. Determination of area or perimeter quadrilateral, and any quadrilateral, 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 - coverings for floors, walls or ceilings, as well as for laying out lawns and beds, etc.

Instructions

1. To determine 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 enough to be squared. Thus, the area of ​​a square is equal to the length of its side squared. The unit of measurement for area can be square millimeters, centimeters, decimeters, meters, kilometers. To determine the area of ​​a square, you can use the formula S = aa, where S – area of ​​the square, a- side of a square.

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

3. The perimeter is the total length of the shape'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 = a + a + a + a or P = 4a, where P is the perimeter, a 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 baseboards if the size of one side of the room is 6 meters. Write down the formula P = 4a. Substitute the data specified in the condition into it: P (rooms) = 4 x 6 = 24 meters. Consequently, the length of the ceiling plinths will also be equal to 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 type of rhombus in which all of the angles are equal to 90 degrees. Being a positive quadrilateral, a circle can be described or inscribed around a square. The radius of a circle inscribed in a square can be found using the formula: R = t/2, where t is the side of the square. If the circle is circumscribed around it, then its radius is found as follows: R = (?2*t)/2 Based on these formulas, it is possible derive new ones to find the perimeter of a square: P = 8*R, where R is the radius of the inscribed circle; P = 4*?2*R, where R is the radius of the inscribed circle. The square is unique geometric figure, from the fact that it is unconditionally symmetrical, regardless of how and where to draw the axis of symmetry.