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.
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
- Alexandrova E.I. Mathematics. 2nd grade. - M.: Bustard, 2004.
- Bashmakov M.I., Nefedova M.G. Mathematics. 2nd grade. - M.: Astrel, 2006.
- Dorofeev G.V., Mirakova T.I. Mathematics. 2nd grade. - M.: Education, 2012.
- Festival.1september.ru ().
- Nsportal.ru ().
- Math-prosto.ru ().
Homework
- Find the perimeter of a rectangle whose length is 13 meters and width is 7 meters.
- Find the semi-perimeter of a rectangle if its length is 8 cm and width is 4 cm.
- Find the perimeter of a rectangle if its semi-perimeter is 21 dm.
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.
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%.
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:
- 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.
- 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.
- 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:
- Small objects are measured in cm², such as photographs, textbook covers, and sheets of paper.
- In dm² you can measure a geographical map, window glass, a painting.
- 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:
- 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.
- 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.
- 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. 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.
Below in the article you will learn what it 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 another interesting construction application problem.A little theory:
Perimeter is the length of a geometric figure along its outer boundary.
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 summarize! In order to calculate the perimeter of a rectangle, you need to add up all its sides.
Typical mathematical and practical problems:
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 the rectangle is = 2 * (5 + 10) = 30 cm.
Answer: 30 cm.
Task #2:
Input: Determine the sides of the rectangle expressed in integers if the perimeter of the rectangle is 10.
Solution:
Using the formula, we determine the sum of the lengths of the sides (a + b) = P / 2 = 10 / 2 = 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.
Problem No. 3 (practical):
Initial data: Determine the number of skirting boards sufficient 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 construction stores, skirting boards are sold not by linear meters, but by the piece. Therefore, we accept the following integer. That'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 very important practical significance, for example in construction or general planning of a territory.
This page presents the simplest online calculator for calculating the perimeter of a rectangle. With this program you can find the perimeter of a rectangle in one click if its length and width are known.
