What is a perimeter? How to find the perimeter? Perimeter of a square and a rectangle. Methods for determining and examples of solutions How to calculate the perimeter of a figure

Today we will talk about how to calculate polygon perimeter. But first, let's talk about the variety of figures. Look at the picture. What figures do we see here? These are a rectangle and a square - polygons that have four sides, as well as a triangle with three sides, and a pentagon with five sides.

And how to find the perimeter of these figures?

To find the perimeter of a polygon, add the lengths of all its sides..

The perimeter is indicated by a capital Latin letter R.

Let's look at a few examples.

Calculate the perimeter of the polygon O. As we said earlier, the perimeter of a polygon is the sum of the lengths of all its sides. Let's add all the sides of our polygon:

P \u003d 15 + 17 + 10 + 10 + 20 + 15 \u003d 87

But you can calculate the perimeter in another way, using multiplication. We see that some sides of the polygon are the same. We have two sides of 15 conventional units and two more of 10. Let's write the expression:

P \u003d 15 × 2 + 10 × 2 + 17 + 20 \u003d 87

Now let's talk about the features of calculating the perimeter of some polygons.

A rectangle is a quadrilateral whose opposite sides are equal. For example, to calculate A with sides a and b, you need to add these sides and multiply the result by 2:

P(rectangle) = (a + b) × 2

That is, if the side of the rectangle a \u003d 5 cm, and the side of the rectangle b \u003d 3 cm, then the perimeter of the rectangle will be:

P \u003d (5 + 3) × 2 \u003d 16 cm

But how to find the unknown sides of a rectangle if its perimeter and only one of the sides are known?

P(rectangle) = 2 × a + 2 × b

a \u003d (P - 2 × b) ÷ 2 or b \u003d (P - 2 × a) ÷ 2

Example: The perimeter of a rectangle is 16 cm, side a = 5 cm. What are the other sides of the rectangle?

If we know one side of a rectangle, then the lengths of two of the four sides are known to us. Let's find the other two sides. That is, we find one, and the second will be equal to it.

side b \u003d (16 - 2 × 5) ÷ 2 \u003d 3 cm

Answer: A rectangle has two sides of 5 cm and two of 3 cm.

A square is a rectangle with all sides equal. To calculate, you need to multiply the length of one side by 4:

P(square) = a × 4

For example, square B has side a = 5 cm. To find its perimeter:

P (B) \u003d 5 × 4 \u003d 20 cm

And if the perimeter of a square is known, how to find the lengths of its sides? Very simply, you need to divide its perimeter into four:

a = P ÷ 4

Example: The perimeter of a square is 24 cm. What are its sides?

a = 24 ÷ 4 = 6

Answer: The sides of a square are 6 cm.

In the similarity of calculating the perimeter of a square, the perimeter of all equilateral polygons. That is, it is equal to the length of one of its sides multiplied by the number of sides.

If the length of one side of the polygon is a, and the number of its sides is n, then its perimeter will be equal to:

P(equilateral polygon) = a × n

For example, a pentagon D has side a = 6 cm. Let's find its perimeter:

R (D) \u003d 6 × 5 \u003d 30 cm

Well, if the perimeter of an equilateral polygon is known, then calculating the lengths of its sides is very simple, you need to divide its perimeter by the number of sides.

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

• To calculate the perimeter of 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 “P” sign in order to know whose perimeter you are finding.
• The perimeter is measured in units of length: mm, cm, m, km, etc.

Distinctive features of the 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
• 2 way. Add the width and length, and multiply by 2. P = (a + b) 2. OR P \u003d 2 a + 2 b. The sides of a rectangle that lie opposite each other (opposite) are called the length and width.

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

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

An example of a problem for calculating the perimeter of a rectangle:

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

Memorize the formulas for calculating the perimeter of a rectangle!

Semiperimeter is the sum of one length and one width .

• Semiperimeter of a rectangle - when you perform the first action in brackets - (a+b).
• To get the perimeter from the 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 in square units (mm 2, cm 2, m 2, km 2, etc.)

Class: 2

Target: Learn how to find the perimeter of a rectangle.

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

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

DURING THE CLASSES

1. Organizational moment

Check readiness for the lesson. Greetings.

The lesson starts
He will go to the guys for the future.
Try to understand everything -
And count carefully.

2. Mental account

a) The use of magical figures. ( Appendix 1 )

- Let's fill in the cells of the magic square, name its features (the sum of the numbers along the horizontals, verticals and diagonals are equal) and determine the magic number. (39)

In a chain, children fill out a square on the board and in notebooks.

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

- The sums of the numbers in the corners that form the triangle are equal. Let's find the magic numbers in the triangle. Find the missing number. Mark it on the whiteboard.

3. Preparation for learning new material

- Before you geometric shapes. Name them in one word. (Quadangles).
- Divide them into 2 groups. ( Annex 3 )
What are rectangles. (Rectangles are quadrangles with all right angles.)
What can be learned by knowing the lengths of the sides of quadrilaterals? The 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 rectangles not known for all sides?
What are the properties of opposite sides of rectangles? (A rectangle has opposite sides equal.)
If opposite sides are equal, should all sides be measured? (No.)
- That's right, just measure the length and width.
- How to calculate in a convenient way? (Students work orally with comments.)

4. Explore 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 in.

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

How to write it differently?

P = a + a + in + in,
P = a x 2+ in x 2,
R = ( a + in) x 2.

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

5. Fixing

Page 44 no. 2.

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

6. Physical Minute. signal cards

How many green cells
So many slopes.
We clap our hands so many times.
We stomp our feet so many times.
How many circles do we have here
So many jumps.
We will swear so many times
So let's pull up now.

7. Practical work

- You have geometric figures in envelopes on your desks. How shall we name 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 check with a 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.

R 1 \u003d (2 + 6) x 2 \u003d 16 R 1 \u003d 2 x 2 + 6 x 2 \u003d 16
R 1 \u003d 2 + 2 + 6 + 6 \u003d 16
R 2 \u003d 3 + 3 + 5 + 5 \u003d 16 R 2 \u003d (3 + 5) x 2 \u003d 16
R 3 \u003d 4 + 4 + 4 + 4 \u003d 16 R 4 \u003d 1 + 1 + 7 + 7 \u003d 16

8. Graphic dictation

Left 6 cells. They made a point. We start moving. 2 - right, 4 - right down, 10 - left, 4 - right up. What figure? Turn it into a rectangle. Complete. Find R in different ways.

P \u003d (5 + 2) x 2 \u003d 14.
P \u003d 5 + 5 + 2 + 2 \u003d 14.
P \u003d 5 x 2 + 2 x 2 \u003d 14.

9. Finger gymnastics

They multiplied, they multiplied.
We are very, very tired.
We will interlace our fingers and connect our palms.
And then, as soon as we can, we squeeze it tightly.
There is a lock on the doors.
Who couldn't open it?
We knocked on 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 by condition(Appendix 8 )

Rectangle length - 12 dm
Width - 3 dm m.
R - ?
In the first step, we find the width: 12 - 3 \u003d 9 (dm) - width
Knowing the length and width, we find out P in one of the ways.
P \u003d (12 + 9) x 2 \u003d 42 dm

11. Independent work

12. Summary of the lesson

- What did you learn. How was the P of a rectangle found?

13. Evaluation

Students' answers are evaluated at the blackboard and selectively in the process of independent work.

14. Homework

S. 44 No. 5 (with explanations).

The ability to find the perimeter of a rectangle is very important for solving many geometric problems. Below is a detailed instruction on finding the perimeter of different rectangles.

How to find the perimeter of a regular rectangle

A regular rectangle is a quadrilateral whose parallel sides are equal and all angles = 90º. There are 2 ways to find its perimeter:

Add up all sides.

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

Solution (sequence of actions and reasoning):

• Since we know the width and length of the rectangle, finding its perimeter is not difficult. The width is parallel to the width, and the length is the length. Thus, in a regular rectangle, there are 2 widths and 2 lengths.
• Add up all sides (3 + 3 + 6 + 6) = 18 cm.

Answer: P = 18 cm.

The second way is as follows:

You need to add the width and length, and multiply by 2. The formula for this method is as follows: 2 × (a + b), where a is the width, b is the length.

As part of this task, we get the following solution:

2x(3 + 6) = 2x9 = 18.

Answer: P = 18.

How to find the perimeter of a rectangle - square

A square is a regular quadrilateral. Correct because all its sides and angles are equal. There are two ways to find its perimeter:

• Add up all of its sides.
• Multiply its side by 4.

Example: Find the perimeter of a square if its side = 5 cm.

Since we know the side of the square, we can find its perimeter.

Add up all sides: 5 + 5 + 5 + 5 = 20.

Answer: P = 20 cm.

Multiply the side of the square by 4 (because everyone is equal): 4x5 = 20.

Answer: P = 20 cm.

How to Find the Perimeter of a Rectangle - Online Resources

While the steps above are easy to understand and master, there are several online calculators that can help you calculate the perimeters (area, volume) of different shapes. Just type in the required values ​​and the mini-program will calculate the perimeter of the shape you need. Below is a short list.

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

Additional materials
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Teaching aids and simulators in the online store "Integral" for grade 3
Simulator for grade 3 "Rules and exercises in mathematics"
Electronic textbook for grade 3 "Mathematics in 10 minutes"

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.

Example.

It reads like this: quadrilateral ABCD; square EFGH.

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

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)

Decision:
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

The formula for calculating the perimeter of a square

P ABCD=2*(AB+BC)

P ABCD=4*AB

Decision.
1. Draw a square ABCD with the original data.

2. Recall the formula for calculating the perimeter of a square:

P ABCD=4*AB

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?

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?

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

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

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

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

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.