What does the perimeter of a rectangle mean? Perimeter and area of ​​a rectangle. Distinctive features of a rectangle

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).

Task No. 2: “How to find the perimeter 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 width of the pool is 10 m?”

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 the swimmer has covered, 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 such as photographs, textbook covers, and sheets of paper are measured in cm².
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 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. By eye. 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, for example, in m², are compared.

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.

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 different 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.

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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 ​​a 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 the sum of the lengths of all sides of the polygon.

• To calculate the perimeter geometric shapes special formulas are used, where the perimeter is denoted by the letter “P”. It is recommended to write the name of the figure in small letters under the sign “P” so that you know whose perimeter you are finding.
• The perimeter is measured in units of length: mm, cm, m, km, etc.

Distinctive features of a rectangle

• A rectangle is a quadrilateral.
• All parallel sides are equal
• All angles = 90º.
• For example in everyday life a rectangle can be found in the form of a book, monitor, table cover or door.

How to calculate the perimeter of a rectangle

There are 2 ways to find it:

• 1 way. Add up all sides. P = a + a + b + b
• Method 2. Add the width and length and multiply by 2. P = (a + b) 2. OR P = 2 a + 2 b. The sides of a rectangle that lie opposite each other (opposite) are called length and width.

"a"- the length of a rectangle, the longer pair of its sides.

"b"- the width of the rectangle, the shorter pair of its sides.

An example of a problem to calculate the perimeter of a rectangle:

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

Remember the formulas for calculating the perimeter of a rectangle!

Semiperimeter is the sum of one length and one width .

• Semi-perimeter of a rectangle - when you perform the first action in brackets - (a+b).
• To obtain a perimeter from a semi-perimeter, you need to increase it by 2 times, i.e. multiply by 2.

How to find the area of ​​a rectangle

Rectangle area formula S= a*b

If the length of one side and the length of the diagonal are known in the condition, then the area can be found using the Pythagorean theorem in such problems; it allows you to find the length of the side of a right triangle if the lengths of the other two sides are known.

• : a 2 + b 2 = c 2, where a and b are the sides of the triangle, and c is the hypotenuse, the longest side.

Remember!

1. All squares are rectangles, but not all rectangles are squares. Because:
   • Rectangle is a quadrilateral with all right angles.
   • Square- a rectangle with all sides equal.
2. If you find the area, the answer will always be square units(mm 2, cm 2, m 2, km 2, etc.)

When solving, it is necessary to take into account that solving the problem of finding the area of ​​a rectangle only from the length of its sides it is forbidden.

This is easy to verify. Let the perimeter of the rectangle be 20 cm. This will be true if its sides are 1 and 9, 2 and 8, 3 and 7 cm. All these three rectangles will have the same perimeter, equal to twenty centimeters. (1 + 9) * 2 = 20 is exactly the same as (2 + 8) * 2 = 20 cm.
As you can see, we can select endless number of options the dimensions of the sides of the rectangle, the perimeter of which will be equal to the specified value.

The area of ​​rectangles with a given perimeter of 20 cm, but with different sides, will be different. For the example given - 9, 16 and 21 square centimeters, respectively.
S 1 = 1 * 9 = 9 cm 2
S 2 = 2 * 8 = 16 cm 2
S 3 = 3 * 7 = 21 cm 2
As you can see, there are an infinite number of options for the area of ​​a figure for a given perimeter.

Note for the curious. In the case of a rectangle with a given perimeter, the maximum area will be a square.

Thus, in order to calculate the area of ​​a rectangle from its perimeter, you must know either the ratio of its sides or the length of one of them. The only figure that has an unambiguous dependence of its area on its perimeter is a circle. Only for circle and a possible solution.

In this lesson:

• Problem 4. Changing the length of the sides while maintaining the area of ​​the rectangle

Problem 1. Find the sides of a rectangle from the area

The perimeter of the rectangle is 32 centimeters, and the sum of the areas of the squares built on each of its sides is 260 square centimeters. Find the sides of the rectangle.
Solution.

2(x+y)=32
According to the conditions of the problem, the sum of the areas of the squares constructed on each of its sides (four squares, respectively) will be equal to
2x 2 +2y 2 =260
x+y=16
x=16-y
2(16-y) 2 +2y 2 =260
2(256-32y+y 2)+2y 2 =260
512-64y+4y 2 -260=0
4y 2 -64y+252=0
D=4096-16x252=64
x 1 =9
x 2 =7
Now let’s take into account that based on the fact that x+y=16 (see above) at x=9, then y=7 and vice versa, if x=7, then y=9
Answer: The sides of the rectangle are 7 and 9 centimeters

Problem 2. Find the sides of a rectangle from the perimeter

The perimeter of the rectangle is 26 cm, and the sum of the areas of the squares built on its two adjacent sides is 89 square meters. cm. Find the sides of the rectangle.
Solution.
Let's denote the sides of the rectangle as x and y.
Then the perimeter of the rectangle is:
2(x+y)=26
The sum of the areas of the squares built on each of its sides (there are two squares, respectively, and these are squares of width and height, since the sides are adjacent) will be equal to
x 2 +y 2 =89
We solve the resulting system of equations. From the first equation we deduce that
x+y=13
y=13-y
Now we perform a substitution in the second equation, replacing x with its equivalent.
(13-y) 2 +y 2 =89
169-26y+y 2 +y 2 -89=0
2y 2 -26y+80=0
We solve the resulting quadratic equation.
D=676-640=36
x 1 =5
x 2 =8
Now let's take into account that based on the fact that x+y=13 (see above) at x=5, then y=8 and vice versa, if x=8, then y=5
Answer: 5 and 8 cm

Problem 3. Find the area of ​​a rectangle from the proportion of its sides

Find the area of ​​a rectangle if its perimeter is 26 cm and its sides are proportional as 2 to 3.

Solution.
Let us denote the sides of the rectangle by the proportionality coefficient x.
Hence the length of one side will be equal to 2x, the other - 3x.

Then:
2(2x+3x)=26
2x+3x=13
5x=13
x=13/5
Now, based on the data obtained, we determine the area of ​​the rectangle:
2x*3x=2*13/5*3*13/5=40.56 cm 2

Problem 4. Changing the length of the sides while maintaining the area of ​​the rectangle

The length of the rectangle is increased by 25%. By what percentage should the width be reduced so that its area does not change?

Solution.
The area of ​​the rectangle is
S = ab

In our case, one of the factors increased by 25%, which means a 2 = 1.25a. So the new area of ​​the rectangle should be equal to
S2 = 1.25ab

Thus, in order to return the area of ​​the rectangle to the initial value, then
S2 = S/1.25
S2 = 1.25ab / 1.25

Since the new size a cannot be changed, then
S 2 = (1.25a) b / 1.25

1 / 1,25 = 0,8
Thus, the value of the second side must be reduced by (1 - 0.8) * 100% = 20%

Answer: width should be reduced by 20%.