How the perimeter of the rectangle is written. What is the perimeter and its use in practice

The rectangle possesses many distinctive features, based on which the rules for calculating its various numeric characteristics have been developed. So rectangle:

Flat geometric figure;
Quadrangle;
The figure in which the opposite directions are equal and parallel, all the corners are straight.

Perimeter is the total length of all sides of the shape.

Calculation of the perimeter of the rectangle is a pretty simple task.

All you need to know is the width and length of the rectangle. Since the rectangle has two equal lengths and two equal widths, only one side is measured.

The perimeter of the rectangle is equal to the double sum of the 2nd sides of the length and width.

P \u003d (a + b) 2, where a is the length of the rectangle, B is the width of the rectangle.

Also, the perimeter of the rectangle can be found using the sum of all sides.

P \u003d a + a + b + b, where the length of the rectangle, B is the width of the rectangle.

The perimeter of the square is the length of the side of the square, multiplied by 4.

P \u003d A 4, where A is the length of the side of the square.

Supplement: Finding Find Square and Perimeter Rectangles

The study program for grade 3 provides for the study of polygons and their features. In order to understand how to find the perimeter of the rectangle and the area, we will understand what is meant under these concepts.

Basic concepts

Being perimeter and square requires knowledge of some terms. These include:

Right angle. It is formed from 2 rays that have a common principle as a point. When finding out the figures (grade 3), the straight angle is determined by the coal.
Rectangle. This is a quadrangle, all the angles of which are straight. Its parties are called long and width. As you know, the opposite sides of this figure are equal.
Square. It is a quadrangle, all sides of which are equal.

When familiarizing with polygons, their vertices may be called ABSD. In mathematics, it is customary to refer point in the drawings of the letters of the Latin alphabet. The title of the polygon lists all the vertices without skips, for example, the ABC triangle.

Calculation of perimeter

The perimeter of the polygon is the sum of all its sides. This value is indicated by the Latin letter P. The level of knowledge for the proposed examples is 3 class.

Task # 1: "Instruct a 3 cm rectangle and 4 cm long with ABCD vertices. Find the perimeter of the ABCD rectangle.

The formula will look like this: P \u003d AB + BC + CD + AD or P \u003d AB × 2 + BC × 2.

Answer: P \u003d 3 + 4 + 3 + 4 \u003d 14 (cm) or p \u003d 3 × 2 + 4 × 2 \u003d 14 (cm).

Task # 2: "How to find the perimeter of the ABC rectangular triangle if the sides values \u200b\u200bare 5, 4 and 3 cm?".

Answer: P \u003d 5 + 4 + 3 \u003d 12 (cm).

Task # 3: "Find the perimeter of the rectangle, one side of which is 7 cm, and another 2 cm is longer."

Answer: P \u003d 7 + 9 + 7 + 9 \u003d 32 (cm).

Task # 4: "Swimming competitions took place in the pool, the perimeter of which is 120 m. How many meters flew the participant of the competition, if the basin width is 10 m?".

This task is the question of how to find the length of the pool. To solve, find the lengths of the sides of the rectangle. The width is known. The sum of the lengths of the two unknown parties should be 100 m. 120-10 × 2 \u003d 100. To find out the distance that the swimmer overcomed, you need to divide the result obtained by 2. 100: 2 \u003d 50.

Answer: 50 (m).

Calculation of Square

A more complex value is the area of \u200b\u200bthe figure. For its measurement use measurements. The standard among the measurements are squares.

Square Square Side 1 cm is 1 cm². The square decimeter is indicated as dm², and the square meter is m².

Application areas of units may be:

Smalls measure small objects, such as photos, textbooks, sheets of paper.
In DM² you can measure the geographic map, window glass, picture.
For measuring gender, apartments, land use m².

If you draw a 3 cm rectangle with a length and 1 cm wide and split into squares with a side of 1 cm, then 3 squares fit in it, and therefore its area will be 3 cm². If the rectangle is smashed into squares, we find the perimeter of the rectangle also without difficulty. In this case, it is 8 cm.

Another way to calculate the number of squares accompanied in the figure is the use of a palette. Draw a square with an area of \u200b\u200b1 dm² on a tank, which is 100 cm². Let's place tracker on the figure and consider the number of square centimeters in one row. After that, we find out the number of rows, and then moving the values. So, the area of \u200b\u200bthe rectangle is a product of its length and width.

Methods of comparing areas:

Approximately. Sometimes it is enough just to look at items, because in some cases the naked eye shows that one figure takes more space, such as a textbook lying on the table next to the penalty.
Overlay. If the figures coincide when applied, their square is equal. If one of them is fully placed inside the second, then its area is less. Places occupied by a notebook sheet and a page from the textbook can be compared by imposing them on each other.
By the number of measure. Figures when applied may not coincide, but have the same area. In this case, you can calculate the number of squares to which the figure is broken.
Numbers. Compared numerical values \u200b\u200bmeasured by the same measurement, for example, in m².

Example №1: "The seamstress sewed a baby blanket from square multicolored flaps. One alcohol 1 dm long, in a row of 5 pieces. How many decimamers of the tape will need a seam for the processing of the edges of the blanket, if the area is known 50 dm²? ".

To solve the task, you need to answer the question how to find the length of the rectangle. Next, we find the perimeter of the rectangle composed of squares. It is clear from the task that the width of the blanket is 5 dm, we calculate the length, separating 50 to 5, and we get 10 dm. Now find the perimeter of the rectangle with the sides 5 and 10. p \u003d 5 + 5 + 10 + 10 \u003d 30.

Answer: 30 (m).

Example №2: "The site discovered on the excavations where the ancient treasures may be. How many territory will have to be explored by scientists if the perimeter is known 18 m and the width of the rectangle is 3 m? ".

We define the length of the site by doing 2 actions. 18-3 × 2 \u003d 12. 12: 2 \u003d 6. The desired territory will also be 18 m² (6 × 3 \u003d 18).

Answer: 18 (m²).

Thus, knowing the formulas, calculate the area and the perimeter will not be difficult, and the above examples will help to practice in solving mathematical problems.

In this lesson, we will get acquainted with a new concept - the perimeter of the rectangle. We will formulate the definition of this concept, withdraw the formula to calculate it. We will also repeat the combination law of addition and the distribution law of multiplication.

In this lesson, we will get acquainted with the perimeter of the rectangle and its calculation.

Consider the following geometric shape (Fig. 1):

Fig. 1. Rectangle

This figure is a rectangle. Recall what distinguishing features of the rectangle we know.

The rectangle is a quadrangle, who has four straight corners and partially equal side.

What in our life can have a rectangular shape? For example, a book, a table cover or a land plot.

Consider the following task:

Task 1 (Fig. 2)

Around the land plot, it was necessary to put a fence. The width of this plot is 5 meters, length - 10 meters. The fence is what length will be from builders?

Fig. 2. Illustration for task 1

The fence put on the borders of the site, so to find out the length of the fence, you need to know the length of each part. This rectangle has the parties: 5 meters, 10 meters, 5 meters, 10 meters. We will make an expression to calculate the length of the fence: 5 + 10 + 5 + 10. We use the prolonged law of addition: 5 + 10 + 5 + 10 \u003d 5 + 5 + 10 + 10. In this expression there are sums of the same terms (5 + 5 and 10 + 10). We replace the sum of the same terms of the works: 5 + 5 + 10 + 10 \u003d 5 · 2 + 10 · 2. Now we use the distribution law of multiplication relative to addition: 5 · 2 + 10 · 2 \u003d (5 + 10) · 2.

Find the value of the expression (5 + 10) · 2. First, perform an action in brackets: 5 + 10 \u003d 15. And then repeat the number 15 twice: 15 · 2 \u003d 30.

Answer: 30 meters.

Perimeter of rectangle - The sum of all his sides. Formula for counting perimeter rectangle:, Here a is the length of the rectangle, and B is the width of the rectangle. The sum of length and width is called half-reader. In order to get a perimeter from half aimetime, you need to increase it 2 times, that is, multiply by 2.

We use the rectangle perimeter formula and we will find the perimeter of the rectangle with the sides of 7 cm and 3 cm: (7 + 3) · 2 \u003d 20 (cm).

The perimeter of any figure is measured in linear units.

In this lesson, we became acquainted with the perimeter of the rectangle and the formula for its calculation.

The number of numbers and the amount of numbers is equal to the amount of products of a given number and each of the terms.

If the perimeter is the sum of the lengths of all sides of the shape, then the half-meter is the sum of the same length and one width. We find a half-meter, when we work according to the formula of finding the perimeter of the rectangle (when we perform the first action in brackets - (a + b)).

List of references

Alexandrova E.I. Mathematics. Grade 2. - M.: Drop, 2004.
Bashmakov M.I., Nefodeova M.G. Mathematics. Grade 2. - M.: Astrel, 2006.
Dorofeev G.V., Miraca T.I. Mathematics. Grade 2. - M.: Enlightenment, 2012.

Festival.1September.ru ().
Nsportal.ru ().
Math-Prosto.ru ().

Homework

Find a rectangle perimeter, which has a length of 13 meters, and the width is 7 meters.
Find a semi-meter of a rectangle if its length is 8 cm, and the width is 4 cm.
Find the perimeter of the rectangle if its half-version is 21 dm.

Perimeter- This is the sum of the lengths of all sides of the polygon.

To calculate the perimeter of geometric shapes, special formulas are used, where the perimeter is indicated by the letter "P". The name of the figure is recommended to write with small letters under the sign "P" to know whose perimeter you find.
The perimeter is measured in units of length: mm, cm, m, km, etc.

Distinctive features of the rectangle

Rectangle is a quadrangle.
All parallel sides are equal
All angles \u003d 90º.
For example, in everyday life, a rectangle can occur in the form of books, monitor, covers from a table or door.

How to calculate the perimeter of the rectangle

There are 2 ways to find:

1 way. We fold all sides. P \u003d a + a + b + b
2 way. Fold the width and length, and multiply by 2. P \u003d (A + B) · 2.OR P \u003d 2 · a + 2 · b.The sides of the rectangle, which lie against each other (opposite), are called long and width.

"A" - The length of the rectangle, a longer pair of his sides.

"B" - The width of the rectangle, a shorter pair of his sides.

An example of a task to count the perimeter of the rectangle:

Calculate the perimeter of the rectangle, there is its width equal to 3 cm, and the length is 6.

Remember the formula for calculating the perimeter of the rectangle!

Half-meter- this is the sum of the same length and one width .

Half-meter rectangle -when you perform the first action in brackets - (A + B).
In order to get a perimeter from half aimetimeter, you need to increase it by 2 times, i.e. Multiply by 2.

How to find a rectangle area

The formula of the square of the rectangle S \u003d A * B

If the condition is known for the length of one side and the length of the diagonal, then the area can be found using the Pythagora theorem, it allows you to find the length of the side of the rectangular triangle if the length of two other sides is known.

: a 2 + b 2 \u003d C 2, where a and b is the side of the triangle, and C is hypotenuse, the longest side.

Remember!

All squares are rectangles, but not all rectangles are squares. Because:
- Rectangle - This is a quadrilateral with all straight corners.
- Square - Rectangle, whose all parties are equal.
If you find the square, the answer will always be in square units (mm 2, cm 2, m 2, km 2, etc.)

Often, on the Internet, you can find ridicule on how knowledge in mathematics - integrals, differentials, trigonometric functions and other sections of the subject - do not help relieve human life. Such jokes are in vain, because how it helps the ability to correctly calculate the perimeter of the square, rectangle and other geometric shapes in construction work. Flow consumption: tiles, wallpaper, flooring - not to determine without an understanding of elementary mathematical formulas and geometric shapes.

Properties of Square

Any calculations in mathematics are based on object properties. To answer the question: "What is the perimeter of the square?" - It is recommended to remember the distinctive characteristics of this figure.

Equality of all sides.
The presence of four angles of 90 degrees.
Parallelism of the sides.
Rotary symmetry. When the shape is rotated, it remains unchanged.
The ability to describe and enter the circle.
Diagonal when crossing each other in half.
The area of \u200b\u200bthe figure characterizes a square-filled place in two-dimensional space.
The perimeter of the figure is nothing but the sum of the lengths of his parties.
From the previous property, it follows that the units of measurement of the perimeter will be units of length: m, cm, dm and others.

To count the plinths to complete the repair in a square room, you need to know the length of the room. To do this, it is necessary to calculate its perimeter.

Perimeter

Translated from the Greek language, the word means "measuring around". The term applies to all closed figures: a square, circle, rectangle, triangle, trapezoids and other things. Knowledge to determine the perimeter of elementary figures is needed to solve complex geometric tasks with objects of incorrect shape. For example, to calculate the plinths in the room layout type "G", or as elsewhere called, "boot", it will take to determine the perimeter of the square and the rectangle. After all, the form of the room consists of these elementary figures.

The generally accepted designation of this value is the letter R. each figure, taking into account its properties inherent in its formula to determine the perimeter.

Properties of rectangles

Equality of opposite sides.
Equality of diagonals.
Ability to describe the circle.
The heights of the rectangle are equal to its parties.
The amount of corners is 360 degrees, and all the corners are direct.
Parallelism of opposite sides.
Perpendicularity of the adjacent parties.
The sum of the squares of the rectangle diagonals is equal to the sum of the squares of its sides.
Crossing, diagonally divide each other in half.
The inability to enter a circle in the figure.

Perimeter Square

Depending on the installed (known) square parameters, there are different formulas to determine its perimeter. The simple task is to calculate the perimeter at the set length of its side (C). In this case, p \u003d C + C + C + C or 4 * s. For example, the length of the sides of the square is 7 cm, then the perimeter of the figure is 28 cm (4 * 7).

In the first case, everything is clear, but how to find the perimeter of the square, knowing his area? And then everything is extremely clear. Since the figure of the figure is determined by multiplying one side to another, and the square of all parties are equal to, it is necessary to extract the root from the known value. Example: There is a square with an area of \u200b\u200b25 dm 2. The root of 25 is 5 - this value characterizes the length of the sides of the square. Now, substituting the value found - 5 dm 2 - to the initial perimeter formula, you can solve the problem. The answer will be a value of 20 dm. That is, 4 multiplied by 5, they received the desired value.

Square and circle

From the properties of the considered figure floats that in the square you can enter the circle and also describe it around the figure.

The first option is to find the perimeter on the radius of the circle described. In addition to the square, the vertices of which are on the circle. The radius of the circle is 1/2 diagonal length. It turns out that the diameter is diagonally. Now it is necessary to consider the rectangular triangle, which turned out as a result of dividing the diagonal of the square. The solution of the problem comes down to finding the parties of this triangle. Sun is a known magnitude, the diameter of the circumference described. Suppose it is 3 cm. Pythagora theorem in the case of equal sides of the triangle, it will look like this: 2C 2 \u003d 3 2. In the formula, the designation C is the length of the side of the triangle and square; 3 is the known magnitude of the hypotenuse. From here, C \u003d √9 / 2. Knowing the side of the square, its perimeter is not a problem.

A feature of the inscribed circle is the division of the sides of the square in half. Therefore, the radius is equal to half the length of the side of the square. Then side C \u003d 2 * radius. The perimeter of the square in this case is 4 * 2 * radius or 8 circle radius.

Perimeter of rectangle

The most elementary formula for determining the perimeter of the rectangle through the known values \u200b\u200bof its parties looks like this: p \u003d 2 (a + b), where a and b - the length of the sides of the figure.

The diagonal of the rectangle is similar to the square divides the figure in half, forming a rectangular triangle. However, the task is complicated by the fact that the parties of this triangle are unequal. In the case of the known value of one of the parties and the diagonal, the second one can be found, following the Pythagoree theorem: d 2 \u003d a 2 + to 2, where a and in the parties of the figure, and d - diagonal.

If none of the parties are unknown, then the trigonometry knowledge includes: sinuses, cosines and other functions.

Finding the perimeter according to the described circle and the known diameter is reduced to the fact that the diameter is equal to the length of the diagonal of the figure. Further, the solution of the problem is determined by the presence of known quantities. If the angles are given, then through trigonometric functions. If the side is given, the answer will be found through the Pythagore's theorem.

Rectangle and trigonometric functions

For clarity, an example of solving the problem is given. Danar: Rectangle ABSD; Diagonal length ( d.) 20 cm; angle f. - 30 °. Find perimeter shapes.

From trigonometry It is necessary to recall the following: the sine of the angle in the rectangular triangle is equal to the attitude of the opposite category to the hypotenuse. Sine 30 ° (There are tables for which the values \u200b\u200bof trigonometric functions can be determined for the correct angles) is 1/2. It turns out 1/2 \u003d relation to d.. Unknown value in will be equal d./ 2 \u003d 20/2 \u003d 10 cm.

To calculate the perimeter, you should find the second side of the shape. It is possible through the theorem of the Pythagora, since the length of hypotenuses and one of the cathettes are known or again through the attitude of the parties for the cosine of the angle.

Cosine corner f. It is expressed as the ratio of the adjacent catech for hypotenuse and is √3 / 2.

√3/2=n / D., n \u003d (D* √3) / 2 or 10 * √3. After removing the root of 3, we obtain the length of the side of the triangle: 10 * 1.73 \u003d 17.3 cm.

Perimeter is 2 (17.3 + 10) \u003d 2 * 27.3 \u003d 54.6 cm.

Perimeter and side attitude

The school program meets the geometry challenges, when the lengths of the sides of the rectangle are expressed by their attitude towards each other. Consideration of solving such a task is presented below.

It is known that the sum of the lengths of all sides of the rectangle, that is, its perimeter is 84 cm. The ratio of length (D) to width (W) - 3: 2. Find the sides of the figure.

Solution: Let the length be 3x, and the width 2x, according to the ratio from the condition of the problem. The formula of the perimeter of the rectangle with the resulting databases of the parties will be as follows: 3x + 3x + 2x + 2x \u003d 84. Next, 10x \u003d 84, x \u003d 8.4 cm. Substitting x in the expression of the length and width of the rectangle, you can find the desired values. Length will be: 3 * 8.4 \u003d 25.2 cm; Width: 2 * 8.4 \u003d 16.8 cm.

The article is devoted to solving the most common tasks in the school program. And this is not all ways to find the perimeter of the square and rectangle.

Class: 2

Goal: Get acquainted with the reception of the perimeter of the rectangle.

Tasks:to form the ability to solve problems associated with finding the perimeter of the figures, develop skills to draw geometric shapes, consolidate the ability to calculate, applying with a moving property of addition, develop an oral account skill, logical thinking, to educate cognitive activity and the ability to work in the team.

Equipment:ICT (multimedia projector, presentation to the lesson), pictures with geometric shapes for fizminutka, model of the magic square, for students - models of geometric shapes, marker boards, rules, textbooks, notebooks.

DURING THE CLASSES

1. Organizational moment

Check availability to the lesson. Greeting.

The lesson begins,
He will go guys in the future.
Try to understand everything -
And carefully read.

2. Oral account

a) The use of magic figures. ( Appendix 1 )

- Fill the cells of the magic square, name it features (the amount of numbers horizontally, verticals and diagonals are equal to) and determine the magic number. (39)

In the chain, children fill the square on the board and in notebooks.

b) acquaintance with the properties of magic triangles. ( Appendix 2. )

- The sums of numbers in the corners forming the triangle are equal. Find the magic numbers in the triangle. Determine the missed number. Mark it on a marker board.

3. Preparation for the study of new material

- Before you geometric shapes. Name them in one word. (Quadriclers).
- divide them into 2 groups. ( Appendix 3. )
- What are rectangles. (Rectangles are quadrangles, who have all the corners direct.)
- What can I know, knowing the length of the sides of the quadricles? Perimeter - the sum of the side of the sides of the figures.
- Find the perimeter of a white figure, yellow.
- Why are the rectangles know not all parties?
- What are the properties of the opposite sides of the rectangles? (At the rectangle the opposite parties are equal).
- If the opposite sides are equal, is it necessary to measure all parties? (No.)
- That's right enough to measure the length and width.
- How to calculate in a convenient way? (Students work orally with commenting.)

4. Study of a new topic

- Read the topic of our lesson: "Rectangle Perimeter". ( Appendix 4. )
- Help find the perimeter of this figure if its length is equal - but, and width - at.

Those who want to find the board. Students in notebooks write down the decision.

- How to write it differently?

P \u003d. but + but + at + at,
P \u003d. but x 2 +. at x 2
P \u003d ( but + at) x 2.

- We got a formula for finding the perimeter of the rectangle. ( Appendix 5. )

5. Fastening

P. 44 № 2.

Children read and write down the condition, the question drawn the figure, find p in different ways, write the answer.

6. Fizminutka. Signal Cards

How many green cells,
So much perform slopes.
So many times with clateers.
So many times with football legs.
How many cramps here are
So much let's jump.
We will come together as many times
So tighten now.

7. Practical work

- On your desks lie in the envelopes geometric shapes. How do we call them?
- What are rectangles?
- What do you know about opposite sides of rectangles?
- Measure the sides of the figures by options, find the perimeter in different ways.
- Check from a neighbor.

Multi-test notebooks.

- Read: How did you find the perimeter? What can be said about the perimeters of these figures? (They are equal).
- Instruct the rectangle with the same P, but other parties.

P 1 \u003d (2 + 6) x 2 \u003d 16 p 1 \u003d 2 x 2 + 6 x 2 \u003d 16
P 1 \u003d 2 + 2 + 6 + 6 \u003d 16
P 2 \u003d 3 + 3 + 5 + 5 \u003d 16 p 2 \u003d (3 + 5) x 2 \u003d 16
P 3 \u003d 4 + 4 + 4 + 4 \u003d 16 p 4 \u003d 1 + 1 + 7 + 7 \u003d 16

8. Graphic dictation

Left 6 cells. Put a point. We start moving. 2 - right, 4 - right down, 10 - left, 4 - right upwards. What figure? Turn it into a rectangle. Full. Find p in different ways.

P \u003d (5 + 2) x 2 \u003d 14.
P \u003d 5 + 5 + 2 + 2 \u003d 14.
P \u003d 5 x 2 + 2 x 2 \u003d 14.

9. Fingering gymnastics

Moved, multiplied.
Very, we are very tired.
Our fingers we rush and connect the palm.
And then, as soon as we can, firmly squeeze.
On the doors hanging the castle.
Who could not open it?
We knocked the lock,
We are castle castors,
We shouted the lock and opened.

(Words are accompanied by movements)

10. Drawing up and solving the problem of condition(Appendix 8. )

Rectangle length - 12 dm
Width - 3 dm m.
R - ?
In the first action, find the width: 12 - 3 \u003d 9 (dm) - width
Knowing the length and width, we will learn in one of the ways.
P \u003d (12 + 9) x 2 \u003d 42 dm

11. Independent work

12. Total lesson

- What was studied. How did you find a rectangle?

13. Postage

Answers are evaluated by students at the board and selectively in the process of independent work.

14. Easter task

P. 44 No. 5 (with explanations).