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 do without the perimeter formula and the skills to calculate it.
Basic concepts
A rectangle is a quadrilateral in which all angles are right and opposite sides are pairwise equal and parallel. In our life, many figures are in the shape of a rectangle, for example, the surface of a table, a notebook, and so on.
Consider an example: a fence must be placed along the boundaries of the land. In order to find out the length of each side, you need to measure them.
Rice. 1. Land plot in the shape of a rectangle.
The land plot has sides with a length of 2 m, 4 m, 2 m, 4 m. Therefore, in order to find out the total length of the fence, you must add the lengths of all sides:
2+2+4+4= 2 2+4 2 =(2+4) 2 =12 m.
It is this value that is generally called the perimeter. Thus, to find the perimeter, you need to add all the sides of the figure. The letter P is used to designate the perimeter.
To calculate the perimeter of a rectangular figure, you do not need to divide it into rectangles, you need to measure only 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 are 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 system of measurement.
Shape Perimeter Formula
If we take into account the fact that 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 \u200b\u200bthe figure, you can use to express the other side through 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 the sides of a figure, the semi-perimeter is the sum of only one length and width. Multiply the semi-perimeter by 2 to get the perimeter.
Area and perimeter are the 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 have learned how to find the perimeter of a rectangle. And also got acquainted with the formula for its calculation. This topic can be encountered not only when solving mathematical problems, but also in real life.
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The rectangle has many distinctive features, on the basis of which the rules for calculating its various numerical characteristics have been developed. So the rectangle:
Flat geometric figure;
Quadrilateral;
A figure in which opposite sides are equal and parallel, all angles are right.
Perimeter is the total length of all sides of a 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 the 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 2 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.
Addendum: Finding Find Area and Perimeter of Rectangles
The curriculum for grade 3 provides for the study of polygons and their features. In order to understand how to find the perimeter of a rectangle and the area, let's figure out what is meant by these concepts.
Basic concepts
Finding the perimeter and area requires knowledge of some terms. These include:
- Right angle. It is formed from 2 rays having a common origin in the form of a point. When getting acquainted with the figures (grade 3), the right angle is determined using a square.
- Rectangle. It is a quadrilateral with all right angles. Its sides are called length and width. As you know, the opposite sides of this figure are equal.
- Square. It is a quadrilateral with all sides equal.
When introduced to 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 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 grade 3.
Task #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 number 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).
Task number 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).
Task number 4: "Swimming competitions were held in a pool with a perimeter of 120 m. How many meters did the competitor swim if the pool was 10 m wide?"
In this problem, the question is how to find the length of the pool. Find the lengths of the sides of the rectangle to solve. 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 \u200b\u200bthe figure. Measures are used to measure it. The standard among measurements are squares.
The area of a square with a side of 1 cm is 1 cm². The square decimeter is denoted as dm², and square meter- m².
Areas of application of units of measure can be as follows:
- Small objects are measured in cm², such as photographs, textbook covers, sheets of paper.
- In dm² can be measured geographical map, window glass, picture.
- To measure the floor, apartment, land use m².
If you draw a rectangle 3 cm long and 1 cm wide and divide it into squares with a side of 1 cm, then 3 squares will fit in it, which means that 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 on a tracing paper a square with an area of 1 dm², which is 100 cm². Let's place a tracing paper on the figure and count the number of square centimeters in one row. After that, find out the number of rows, and then multiply the values. So the area of a rectangle is the product of its length and width.
Ways to compare areas:
- Approximately. Sometimes just looking at the objects is enough, because in some cases it can be seen with the naked eye that one figure takes up more space, like, for example, a textbook lying on the table next to the pencil case.
- Overlay. If the figures coincide when superimposed, their areas are equal. If one of them completely fits inside the second, then its area is smaller. The space occupied by a notebook sheet and a page from a textbook can be compared by superimposing them on top of each other.
- 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.
- Numbers. Compare numerical values measured with the same measure, for example, in m².
Example #1: “A seamstress sewed a baby blanket out of square multi-colored shreds. One shred 1 dm long, in a row of 5 pieces. How many decimeters of tape will a seamstress need to finish the edges of a blanket if the area is known to be 50 dm²?
To solve the problem, you need to answer the question of how to find the length of the rectangle. Next, find the perimeter of a rectangle made up of squares. It is clear from the problem that the width of the blanket is 5 dm, we calculate the length by dividing 50 by 5, and we 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 #2: “During the excavation, a site 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?
Determine the length of the section by doing 2 steps. 18-3×2=12. 12:2=6. The desired area will also be equal to 18 m² (6 × 3 = 18).
Answer: 18 (m²).
Thus, knowing the formulas, it will not be difficult to calculate the area and perimeter, and the above examples will help you practice solving mathematical problems.
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 just like (2 + 8) * 2 = 20 cm.
As you can see, we can choose an infinite number of options the dimensions of the sides of the rectangle, the perimeter of which will be equal to the given value.
The area of rectangles with a given perimeter of 20 cm, but with different sides will be different. For the given example - 9, 16 and 21 square centimeters, respectively.
S 1 \u003d 1 * 9 \u003d 9 cm 2
S 2 \u003d 2 * 8 \u003d 16 cm 2
S 3 \u003d 3 * 7 \u003d 21 cm 2
As you can see, there are an infinite number of options for the area of \u200b\u200ba figure with a given perimeter.
Thus, in order to calculate the area of a rectangle from its perimeter, it is necessary to 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 the perimeter is a circle. Only for circle and possibly a solution.
In this lesson:
- Task 4. Change the length of the sides while maintaining the area of the rectangle
Task 1. Find the sides of a rectangle from the area
The perimeter of a 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 condition of the problem, the sum of the areas of the squares built on each of its sides (squares, respectively, four) will be equal to
2x2+2y2=260
x+y=16
x=16-y
2(16-y) 2 +2y 2 =260
2(256-32y+y2)+2y2=260
512-64y+4y 2 -260=0
4y2 -64y+252=0
D=4096-16x252=64
x1=9
x2=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 a rectangle are 7 and 9 centimeters
Task 2. Find the sides of a rectangle from the perimeter
The perimeter of a rectangle is 26 cm, and the sum of the areas of the squares built on its two adjacent sides is 89 square meters. see 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 the squares of the width and height, since the sides are adjacent) will be equal to
x2+y2=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.
(13th) 2 +y 2 =89
169-26y+y 2 +y 2 -89=0
2y2 -26y+80=0
We solve the resulting quadratic equation.
D=676-640=36
x1=5
x2=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
Task 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 the sides are proportional as 2 to 3.
Solution.
Let us denote the sides of the rectangle by the coefficient of proportionality x.
From where 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 cm2
Task 4. Changing the length of the sides while maintaining the area of a rectangle
Rectangle length 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
S 2 \u003d 1.25ab
Thus, in order to return the area of the rectangle to its initial value, then
S2 = S / 1.25
S 2 \u003d 1.25ab / 1.25
Since the new size a cannot be changed, then
S 2 \u003d (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%.
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.
A 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. Area measured square units lengths: 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.
Below in the article you will learn what is and how to find the perimeter of a rectangle if its sides are known. And also how to find the sides of a rectangle if its perimeter is known. And one more interesting construction applied problem.A little theory:
Perimeter is length geometric figure along its outer border.
The perimeter of a rectangle is the sum of the lengths of its sides.
Formulas for calculating the perimeter of a rectangle: P = 2*(a+b) or P = a + a + b + b.
Let's recap! To calculate the perimeter of a rectangle, add up all of its sides.
Typical mathematical and practical tasks:
Task #1:
Initial data: Determine the perimeter of a rectangle with side lengths of 5 cm and 10 cm.
Solution:
According to the formula, the perimeter of a rectangle is = 2 * (5 + 10) = 30 cm.
Answer: 30 cm.
Task #2:
Initial data: Determine the sides of the rectangle expressed as integers, if the perimeter of the rectangle is 10.
Solution:
According to the formula, we determine the sum of the lengths of the sides (a + b) \u003d P / 2 \u003d 10 / 2 \u003d 5
Integer side values can only be 1 + 4 = 5 and 2 + 3 = 5
Answer: The lengths of the sides can only be 2 and 3 or 1 and 4.
Task number 3 (practical):
Initial data: Determine the number of skirting boards in sufficient quantity to repair the floor in a room 5 meters long and 3 meters wide, if the length of one skirting board is 3 meters.
Solution:
Room perimeter = 2 * (5 + 3) = 16 meters
Number of skirting boards = 16 / 3 = 5.33 pieces
Usually in building stores, skirting boards are sold not by linear meters, but by the piece. Therefore, we take the following integer. It's six.
Answer: The number of skirting boards is 6 pieces.
Finally:
Solving the problem of calculating the perimeter is a fairly simple mathematical problem, but it has a very important practical value for example in construction or general planning of the territory.
This page presents the simplest online calculator to calculate the perimeter of a rectangle. With this program, you can find the perimeter of a rectangle in one click if you know its length and width.