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.

Example.

It reads like this: quadrilateral ABCD; square EFGH.

What is the perimeter of a rectangle? Formula for calculating perimeter

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)

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

Formula for calculating the perimeter of a square

P ABCD = 2 * (AB + BC)

P ABCD = 4 * AB

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

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?

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?

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

S AKMO = AK * KM = 7 cm * 2 cm = 14 cm 2.

Answer: 14 cm 2.

Formula for calculating the area of ​​a square

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

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

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

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.

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.

Class: 2

Target: introduce the method of finding the perimeter of a rectangle.

Tasks: develop the ability to solve problems related to finding the perimeter of figures, develop the ability to draw geometric shapes, consolidate the ability to calculate using the commutative property of addition, develop the skill of mental calculation, logical thinking, cultivate cognitive activity and the ability to work in a team.

Equipment: ICT (multimedia projector, presentation for the lesson), pictures with geometric shapes for physical education, a model of a magic square, students have models of geometric shapes, marker boards, rulers, textbooks, notebooks.

DURING THE CLASSES

1. Organizational moment

Checking readiness for the lesson. Greetings.

The lesson begins
It will be useful for the guys.
Try to understand everything -
And count carefully.

2. Oral counting

a) Use of magical figures. ( Annex 1 )

– Fill in the cells of the magic square, name its features (the sum of the numbers along the horizontal, vertical and diagonal lines is equal) and determine the magic number. (39)

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

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

– The sums of the numbers in the angles forming a triangle are equal. Let's find the magic numbers for the triangle. Find the missing number. Mark it on the marker board.

3. Preparing to study new material

– In front of you are geometric shapes. Name them in one word. (Quadrangles).
– Divide them into 2 groups. ( Appendix 3 )
– What are rectangles? (Rectangles are quadrilaterals in which all angles are right.)
– What can you find out by knowing the lengths of the sides of quadrilaterals? Perimeter is the sum of the lengths of the sides of the figures.
– Find the perimeter of the white figure, the yellow one.
– Why are not all sides known for rectangles?
– What are the properties of the opposite sides of rectangles? (A rectangle has equal opposite sides.)
– If opposite sides are equal, is it necessary to measure all sides? (No.)
- That's right, just measure the length and width.
– How to calculate in a convenient way? (Students work orally with commentary.)

4. Study a new topic

– Read the topic of our lesson: “Perimeter of a rectangle.” ( Appendix 4 )
– Help me find the perimeter of this figure if its length is – A, and the width is V.

Those who wish find R at the board. Students write down the solution in their notebooks.

– How can I write this differently?

P = A + A + V + V,
P = A x 2 + V x 2,
P = ( A + V) x 2.

– We have obtained a formula for finding the perimeter of a rectangle. ( Appendix 5 )

5. Consolidation

Page 44 No. 2.

Children read and write down a condition, a question, draw a figure, find P in different ways, and write down the answer.

6. Physical exercise. Signal cards

How many green cells are there?
Let's do so many bends.
Let's clap our hands so many times.
We stamp our feet so many times.
How many circles do we have here?
We'll do so many jumps.
We will sit down so many times
So let's catch up now.

7. Practical work

– On your desks there are geometric shapes in envelopes. What should we call them?
– What are rectangles?
– What do you know about opposite sides of rectangles?
– Measure the sides of the figures according to the options, find the perimeter in different ways.
- We're checking with our neighbor.

Mutual check of notebooks.

– Read: How did you find the perimeter? What can be said about the perimeters of these figures? (They are equal).
– Draw a rectangle with the same P, but different sides.

P 1 = (2 + 6) x 2 = 16 P 1 = 2 x 2 + 6 x 2 = 16
P 1 = 2 + 2 + 6 + 6 = 16
P 2 = 3 + 3 + 5 + 5 = 16 P 2 = (3 + 5) x 2 = 16
Р 3 = 4 + 4 + 4 + 4 = 16 Р 4 = 1 + 1 + 7 + 7 = 16

8. Graphic dictation

There are 6 cells on the left. We've made a point. Let's start moving. 2 – right, 4 – down right, 10 – left, 4 – up right. What figure? Turn it into a rectangle. Finish it. Find R in different ways.

P = (5 + 2) x 2 = 14.
P = 5 + 5 + 2 + 2 = 14.
P = 5 x 2 + 2 x 2 = 14.

9. Finger gymnastics

They multiplied and multiplied.
We are very, very tired.
Let's intertwine our fingers and join our palms.
And then, as soon as we can, we will squeeze it tightly.
There is a lock on the door.
Who couldn't open it?
We knocked the lock
We turned the lock
We twisted the lock and opened it.

(Words are accompanied by movements)

10. Drawing up and solving a problem according to the condition(Appendix 8 )

Rectangle length – 12 dm
Width – 3 dm m.
R - ?
In the first step we find the width: 12 – 3 = 9 (dm) – width
Knowing the length and width, we find out P in one of the following ways.
P = (12 + 9) x 2 = 42 dm

11. Independent work

12. Lesson summary

- What did you learn? How did you find the P of a rectangle?

13.Assessment

Students' answers are assessed at the board and selectively during independent work.

14.Homework

P. 44 No. 5 (with explanations).

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

• To calculate the perimeter of geometric figures, 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 a 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 in square units (mm 2, cm 2, m 2, km 2, etc.)