In this lesson we will introduce a new concept - the perimeter of a rectangle. We will formulate a definition of this concept and derive a formula for its calculation. We will also repeat the combinational law of addition and the distributive law of multiplication.

In this lesson we will learn about the perimeter of a rectangle and its calculation.

Consider the following geometric figure (Fig. 1):

Rice. 1. Rectangle

This figure is a rectangle. Let's remember what distinctive features of a rectangle we know.

A rectangle is a quadrilateral with four right angles and equal sides.

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

Consider the following problem:

Task 1 (Fig. 2)

The builders needed to put up a fence around the plot of land. The width of this section is 5 meters, the length is 10 meters. What length of fence will the builders get?

Rice. 2. Illustration for problem 1

The fence is placed along the boundaries of the site, therefore, to find out the length of the fence, you need to know the length of each side. This rectangle has equal sides: 5 meters, 10 meters, 5 meters, 10 meters. Let's create an expression for calculating the length of the fence: 5+10+5+10. Let's use the commutative law of addition: 5+10+5+10=5+5+10+10. This expression contains sums of identical terms (5+5 and 10+10). Let's replace the sums of identical terms with products: 5+5+10+10=5·2+10·2. Now let's use the distributive law of multiplication relative to addition: 5·2+10·2=(5+10)·2.

Let's find the value of the expression (5+10)·2. First we perform the action in brackets: 5+10=15. And then we repeat the number 15 twice: 15·2=30.

Answer: 30 meters.

Perimeter of a rectangle- the sum of the lengths of all its sides. Formula for calculating the perimeter of a 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 semi-perimeter. To obtain the perimeter from the semi-perimeter, you need to increase it by 2 times, that is, multiply by 2.

Let's use the formula for the perimeter of a rectangle and find the perimeter of a rectangle with sides 7 cm and 3 cm: (7 + 3) 2 = 20 (cm).

The perimeter of any figure is measured in linear units.

In this lesson we learned about the perimeter of a rectangle and the formula for calculating it.

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

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

Bibliography

1. Alexandrova E.I. Mathematics. 2nd grade. - M.: Bustard, 2004.
2. Bashmakov M.I., Nefedova M.G. Mathematics. 2nd grade. - M.: Astrel, 2006.
3. Dorofeev G.V., Mirakova T.I. Mathematics. 2nd grade. - M.: Education, 2012.

1. Festival.1september.ru ().
2. Nsportal.ru ().
3. Math-prosto.ru ().

Homework

1. Find the perimeter of a rectangle whose length is 13 meters and width is 7 meters.
2. Find the semi-perimeter of a rectangle if its length is 8 cm and width is 4 cm.
3. Find the perimeter of a rectangle if its semi-perimeter is 21 dm.

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

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What are rectangle and square

Rectangle is a quadrilateral with all right angles. This means that opposite sides are equal to each other.

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

Quadrangles, including rectangles and squares, are designated 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 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 a 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 rectangle ABCD is denoted as P ABCD, where A, B, C, D are the vertices of the rectangle.

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

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

Example.
Given a rectangle ABCD 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 the original data.
2. Let’s write a formula to calculate the perimeter of a given rectangle:

P ABCD = 2 * (AB + BC)

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

Answer: P ABCD = 16 cm.

Formula for calculating the perimeter of a square

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

P ABCD = 2 * (AB + BC)

Let's use it to determine 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. Let us determine the perimeter of the square.

Solution.
1. Let's draw a square ABCD with the original data.

2. Let us recall the formula for calculating the perimeter of a square:

P ABCD = 4 * AB

3. Let’s substitute our data into the formula:

P ABCD = 4 * 6 cm = 24 cm

Answer: P ABCD = 24 cm.

Problems to find 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 square SEOM 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 plot of land has been 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 excess material for building a fence.

2. Parents decided to renovate the children's room. You need to know the perimeter of the room and its area in order to correctly calculate the amount of wallpaper.
Determine the length and width of the room in which you live. Determine the perimeter of your room.

What is the area of ​​a rectangle?

Square is a numerical characteristic of a figure. 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 a Latin letter S.

To determine 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 the AC by the width of the CM. Let's write this down 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 = AK * KM = 7 cm * 2 cm = 14 cm 2.

Answer: 14 cm 2.

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 the side AB by the width BC, but since they are equal, the result is multiplying the side AB by AB.

S ABCO = AB * BC = AB * AB

Example.
Determine the area of ​​a 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 square

1. Given a rectangle with sides 20 mm and 60 mm. Calculate its area. Write your answer in square centimeters.

2. A dacha plot measuring 20 m by 30 m was purchased. Determine the area of ​​the dacha plot and write the answer in square centimeters.

Perimeter is a geometric term that often appears in problems. To understand what a perimeter is, you should draw an arbitrary polygon and arm yourself with a ruler. Translated from Greek, this term means “I measure around.”

How to calculate perimeter

The perimeter is indicated by a Latin letter P. It can be measured in centimeters, millimeters, meters or decimeters. To find the perimeter, measure the length of all sides of the polygon. The resulting values ​​must be added. The final sum will be the answer to the question: “What is the perimeter of the polygon?”

Perimeter is the length of the lines that limit a closed figure (square, rectangle, triangle, etc.).

For example, in front of you is a polygon with sides of 10, 12, 13 and 11 cm. We add the above numbers (10+12+13+11) and get the sum 46. This is the perimeter of the polygon.

For the convenience of calculating the perimeter in geometry, there are a number of formulas. Each formula corresponds to a specific figure.

Perimeter and area of ​​a square

This is the sum of its four sides. As we know, all sides of a square are equal in size. Therefore, we can find out the perimeter of a square by multiplying its side length by four:

P= a+a+a+a

For example, we have a square with a side of 10 cm.

Answer: 40 cm

P= 10+10+10+10

P=40

Answer: 40 cm

To understand what perimeter and area are, you should understand that perimeter calculates the length of the contour of a figure, and area is the size of its entire surface.

To find out the area of ​​a square, you need to use a simple formula:

S is the area, and is the side of the square.

For example, the problem states that the length of the side of the square is 10 cm.

S= 100cm 2

Answer: 100 cm 2

Perimeter and area of ​​a rectangle

The sides of a rectangle that are opposite each other and have the same length are called opposite. These are length and width, they are conventionally designated by the Latin letters a and b. The formula for calculating the perimeter of a rectangle looks like this:

P= (a+b)*2

Using this formula, we first find the sum of the width and length and then multiply it by two.

For example, we have a rectangle with a length of 6 cm and a width of 2 cm.

P= (6+2) * 2

P= 16

Answer: 16 cm

To find out the area of ​​a rectangle, multiply the length by the width. The formula looks like this:

For example, the task conditions say that the rectangle has a length of 5 cm and a width of 2 cm. We change the letters a and b to the indicated numbers.

S= 5*2

S=10cm 2

Answer: 10 cm 2

Perimeter of a circle (circumference)

Each circle has a center. The distance from the center of the circle to any point located on the circle is called the radius of the circle. Often students confuse the concepts of “circle” and “circle” and try to determine the area of ​​a circle. This is a serious mistake. You should separate the concepts of “circle” and “circle” in your head. A circle does not and cannot have area, it only has length.

To find the perimeter of a circle, you need to calculate its circumference. There is a formula for finding the circumference of a circle:

L = 2πr

L- circumference

π is the number “pi”, a mathematical constant. It is equal to the ratio of the circumference of a circle to the length of its diameter. The ancient name for the number "pi" is Ludolph's number. This number is irrational; its decimal representation after the dot never ends.

π = 3.141 592 653 589 793 238 462 643 383 279 502

For ease of calculation, the value 3.14 is usually used

R is the radius of the circle

D– Circle diameter

So, to determine the perimeter of a circle, we need to find the product of the radius and 2π. If the problem specifies a diameter, then

For example, in front of us is a circle with a radius of 3 cm. Let’s find its perimeter.

L= 2*3,14*3

L=6 π

L=6*3.14

L= 18.84 cm

PTo= 18.84 cm

Answer: 18.84 cm

The difference between perimeter and area

Area is the size of the surface of a figure, and perimeter is the sum of its boundaries.

Area is always measured in square units (cm 2, m 2, mm 2). The perimeter is measured in units of length - centimeters, millimeters, meters, decimeters.

One of the basic concepts of mathematics is the perimeter of a rectangle. There are many problems on this topic, the solution of which cannot be done without the perimeter formula and the skills to calculate it.

Basic Concepts

A rectangle is a quadrilateral in which all the angles are right and the opposite sides are equal and parallel in pairs. In our life, many figures have the shape of a rectangle, for example, the surface of a table, a notebook, etc.

Let's look at an example: A fence must be erected along the boundaries of the land plot. In order to find out the length of each side, you need to measure them.

Rice. 1. A plot of land in the shape of a rectangle.

The plot of land has sides with lengths of 2 m, 4 m, 2 m, 4 m. Therefore, to find out the total length of the fence, you need to add up the lengths of all sides:

2+2+4+4= 2·2+4·2 =(2+4)·2 =12 m.

It is this quantity that is generally called the perimeter. Thus, to find the perimeter, you need to add up all the sides of the figure. The letter P is used to denote the perimeter.

To calculate the perimeter of a rectangular figure, you do not need to divide it into rectangles; you only need to measure all sides of this figure with a ruler (tape measure) and find their sum.

The perimeter of a rectangle is measured in mm, cm, m, km and so on. If necessary, the data in the task is converted into the same measurement system.

The perimeter of a rectangle is measured in various units: mm, cm, m, km and so on. If necessary, the data in the task is converted into one measurement system.

Formula for the perimeter of a figure

If we take into account the fact that the opposite sides of a rectangle are equal, then we can derive the formula for the perimeter of a rectangle:

$P = (a+b) * 2$, where a, b are the sides of the figure.

Rice. 2. Rectangle, with opposite sides marked.

There is another way to find the perimeter. If the task is given only one side and the area of ​​the figure, you can use to express the other side in terms of the area. Then the formula will look like this:

$P = ((2S + 2a2)\over(a))$, where S is the area of ​​the rectangle.

Rice. 3. Rectangle with sides a, b.

Exercise : Calculate the perimeter of a rectangle if its sides are 4 cm and 6 cm.

Solution:

We use the formula $P = (a+b)*2$

$P = (4+6)*2=20 cm$

Thus, the perimeter of the figure is $P = 20 cm$.

Since the perimeter is the sum of all sides of a figure, the semi-perimeter is the sum of only one length and width. To get the perimeter, you need to multiply the semi-perimeter by 2.

Area and perimeter are two basic concepts for measuring any figure. They should not be confused, although they are related. If you increase or decrease the area, then, accordingly, its perimeter will increase or decrease.

What have we learned?

We learned how to find the perimeter of a rectangle. We also got acquainted with the formula for calculating it. This topic can be encountered not only when solving mathematical problems, but also in real life.

Test on the topic

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A rectangle has many distinctive features, based on which rules for calculating its various numerical characteristics have been developed. So, a rectangle:

Flat geometric figure;
Quadrangle;
A figure in which opposite sides are equal and parallel, all angles are right.

The perimeter is the total length of all sides of the figure.

Calculating the perimeter of a rectangle is a fairly simple task.

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

The perimeter of a rectangle is equal to twice the sum of its two sides, length and width.

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

The perimeter of a rectangle can also be found using the sum of all sides.

P= a+a+b+b, where a is the length of the rectangle, b is the width of the rectangle.

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

P = a 4, where a is the length of the side of the square.

Addition: Finding the area and perimeter of rectangles

The curriculum for grade 3 includes the study of polygons and their features. In order to understand how to find the perimeter of a rectangle and area, let's figure out what is meant by these concepts.

Basic Concepts

Finding perimeter and area requires knowledge of some terms. These include:

1. Right angle. It is formed from 2 rays that have a common origin in the form of a point. When learning about shapes (grade 3), a right angle is determined using a square.
2. Rectangle. This is a quadrilateral whose angles are all right. Its sides are called length and width. As you know, opposite sides of this figure are equal.
3. Square. Is a quadrilateral with all sides equal.

When becoming familiar with polygons, their vertices may be called ABCD. In mathematics, it is customary to name points in drawings with letters of the Latin alphabet. The name of the polygon lists all the vertices without gaps, for example, triangle ABC.

Perimeter calculation

The perimeter of a polygon is the sum of the lengths of all its sides. This value is denoted by the Latin letter P. The level of knowledge for the proposed examples is 3rd grade.

Problem #1: “Draw a rectangle 3 cm wide and 4 cm long with vertices ABCD. Find the perimeter of rectangle ABCD."

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

Answer: P=3+4+3+4=14 (cm) or P=3×2 + 4×2=14 (cm).

Problem No. 2: “How to find the perimeter of a right triangle ABC if the sides are 5, 4 and 3 cm?”

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

Problem No. 3: “Find the perimeter of a rectangle, one side of which is 7 cm and the other is 2 cm longer.”

Answer: P=7+9+7+9=32 (cm).

Problem No. 4: “The swimming competition took place in a pool whose perimeter is 120 m. How many meters did the competitor swim if the pool is 10 m wide?”

In this problem the question is 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 sides should be 100 m. 120-10×2=100. To find out the distance covered by the swimmer, you need to divide the result by 2. 100:2=50.

Answer: 50 (m).

Area calculation

A more complex quantity is the area of ​​the figure. Measurements are used to measure it. The standard among measurements is squares.

The area of ​​a square with a side of 1 cm is 1 cm². A square decimeter is denoted as dm², and a square meter is denoted as m².

The areas of application of units of measurement can be:

1. Small objects are measured in cm², such as photographs, textbook covers, and sheets of paper.
2. In dm² you can measure a geographical map, window glass, a painting.
3. To measure a floor, apartment, or plot of land, m² is used.

If you draw a rectangle 3 cm long and 1 cm wide and divide it into squares with a side of 1 cm, then it will fit 3 squares, which means its area will be 3 cm². If the rectangle is divided into squares, we can also find the perimeter of the rectangle without difficulty. In this case it is 8 cm.

Another way to count the number of squares that fit into a shape is to use a palette. Let's draw a square on tracing paper with an area of ​​1 dm², which is 100 cm². Place the tracing paper on the figure and count the number of square centimeters in one row. After this, we find out the number of rows, and then multiply the values. This means that the area of ​​a rectangle is the product of its length and width.

Ways to compare areas:

1. Approximately. Sometimes it is enough just to look at objects, since in some cases it is clear to the naked eye that one figure takes up more space, such as a textbook lying on the table next to a pencil case.
2. Overlay. If the shapes coincide when superimposed, their areas are equal. If one of them fits completely inside the second, then its area is smaller. The spaces occupied by a notebook sheet and a page from a textbook can be compared by superimposing them on top of each other.
3. By the number of measurements. When superimposed, the figures may not coincide, but have the same area. In this case, you can compare by counting the number of squares into which the figure is divided.
4. Numbers. Numerical values ​​measured with the same standard are compared, for example, in m².

Example No. 1: “A seamstress sewed a baby blanket from square multi-colored scraps. One piece 1 dm long, 5 pieces in a row. How many decimeters of tape will a seamstress need to process the edges of a blanket if the area is 50 dm²?”

To solve the problem, you need to answer the question of how to find the length of a rectangle. Next, find the perimeter of a rectangle made up of squares. From the problem it is clear that the width of the blanket is 5 dm; we calculate the length by dividing 50 by 5 and get 10 dm. Now find the perimeter of a rectangle with sides 5 and 10. P=5+5+10+10=30.

Answer: 30 (m).

Example No. 2: “During the excavations, an area was discovered where ancient treasures may be located. How much territory will scientists have to explore if the perimeter is 18 m and the width of the rectangle is 3 m?

Let's determine the length of the section by performing 2 steps. 18-3×2=12. 12:2=6. The required territory will also be equal to 18 m² (6×3=18).

Answer: 18 (m²).

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