What is the squared perimeter. How to find the perimeter of a square if its area is known

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.

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 called the sum of the lengths of all sides of a flat geometric figure, which is measured by 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

  • Because the 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- This 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- This 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.

Many people remember what a square is from a 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 ​​will not be difficult if you take a few minutes to watch this video tutorial explaining the simple rules for making 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.

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.
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. The area is measured in square units of length: 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.
In this example, the area of ​​the 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.

The perimeter of a two-dimensional figure is the total length of its border, equal to the sum of the lengths of the sides of the figure. A square is a figure with four sides of the same length that intersect at an angle of 90°. Since all sides of a square are the same length, it is very easy to calculate its perimeter. This article will tell you how to calculate the perimeter of a square given one side, a given area, and a given radius of a circle circumscribed around the square.

The perimeter is a numerical indicator that is found by the formula 4x, where x is the length of the side of the geometric figure, and 4 is the number of sides of the figure. Let's consider several ways of this calculation.

1st method: Calculating the perimeter on a given side

If the dimensions of the area are known, then from a given value it is possible to find the perimeter of the square. To do this, you need to take the square root, so we find the length of the side, and calculate the final value using the above formula. If you want to find the perimeter of a square along a diagonal line, you will need to use the Pythagorean table.

The geometric figure is divided by a diagonal into isosceles triangles with a right angle, and if the diagonal is known, then the value of the sides of the geometric figure must be calculated using the formula, where the square of z (diagonal) is equal to twice the square of the side u. As a result, we have this value: u is equal to the square root, which was taken from half the square of the hypotenuse. Next, multiply the final value by 4 times and get the perimeter of a geometric figure, that is, a square.

2nd method: Calculating the perimeter from a given area

Formula for calculating the area of ​​a square. The area of ​​any rectangle (and a square is a special case of a rectangle) is equal to the product of its length and its width. Since the length and width of the square are equal, its area is calculated by the formula: A = s*s = s2, where s is the length of the side of the square.

Take the square root of the area value to find the side of the square. To do this, in most cases, use the calculator (enter the area value and press the “√” key). You can also calculate the square root manually.

If the area of ​​a square is 20, then its side is: s = √20 = 4.472.

If the area of ​​a square is 25, then s = √25 = 5.

Multiply the found side by 4 to find the perimeter. Substitute the calculated value of the side into the formula for finding the perimeter: P = 4s. You will find the perimeter of the square.

In our first example: P = 4 * 4.472 = 17.888.

The perimeter of a square whose area is 25 and whose side is 5 is P = 4 * 5 = 20.

3rd way: Calculate the perimeter given the radius of a circle circumscribed around a square

An inscribed square is a square whose vertices lie on a circle.

The ratio 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 the square s, it is necessary to divide the square into 2 right-angled triangles with a diagonal. Each of these triangles will have equal sides a and b and a common hypotenuse c 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 c: a2 + b2 = c2. Since in our case a = b (remember that we are considering a square!), and we know that c = 2r, we can rewrite and simplify this equation:

a2 + a2 = (2r)2″‘; Now let's simplify this equation:

2a2 = 4(r)2; Now we divide both sides of the equation by 2:

(a2) = 2(r)2; Now let's take the square root of both sides of the equation:

a = √(2r). Thus, s = √(2r).

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

Example. Consider a square inscribed in a circle with a radius of 10. This means that the diagonal of the square is 2 * 10 = 20. Using the Pythagorean theorem, we get: 2(a2) = 202, that is, 2a2 = 400. Now we divide both sides of the equation by 2 and we get: a2 \u003d 200. Now we take the square root of both sides of the equation and we get: a \u003d 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 such a method is difficult to remember, so it is better to use the calculation process described above.

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