How to find the perimeter of a figure with different sides. A simple task: how to find the perimeter

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

Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions. All materials are checked by an antivirus program.

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. Means, 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 .

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

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.

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

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. The 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 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.
AT this example the area of a 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 suburban area Write your answer in square centimeters.

In the next test tasks Find the perimeter of the figure shown in the figure.

You can find the perimeter of a shape different ways. You can transform the original shape in such a way that the perimeter of the new shape can be easily calculated (for example, change to a rectangle).

Another solution is to look for the perimeter of the figure directly (as the sum of the lengths of all its sides). But in this case, one cannot rely only on the drawing, but find the lengths of the segments based on the data of the problem.

I want to warn you: in one of the tasks, among the proposed answers, I did not find the one that turned out for me.

c) .

Let's move the sides of the small rectangles from the inner area to the outer one. As a result, the large rectangle is closed. Formula for Finding the Perimeter of a Rectangle

In this case, a=9a, b=3a+a=4a. Thus P=2(9a+4a)=26a. To the perimeter of the large rectangle we add the sum of the lengths of four segments, each of which is equal to 3a. As a result, P=26a+4∙3a= 38a .

c) .

After transferring the inner sides of the small rectangles to the outer area, we get a large rectangle, the perimeter of which is P=2(10x+6x)=32x, and four segments, two of x length, two of 2x length.

Total, P=32x+2∙2x+2∙x= 38x .

?) .

Let's move 6 horizontal "steps" from the inside to the outside. The perimeter of the resulting large rectangle is P=2(6y+8y)=28y. It remains to find the sum of the lengths of the segments inside the rectangle 4y+6∙y=10y. Thus, the perimeter of the figure is P=28y+10y= 38y .

D) .

Let's move the vertical segments from the inner area of the figure to the left, to the outer area. To get a big rectangle, move one of the 4x lengths to the bottom left corner.

We find the perimeter of the original figure as the sum of the perimeter of this large rectangle and the lengths of the remaining three segments P=2(10x+8x)+6x+4x+2x= 48x .

e) .

Transferring inner sides small rectangles to the outer area, we get a large square. Its perimeter is P=4∙10x=40x. To get the perimeter of the original figure, you need to add the sum of the lengths of eight segments, each 3x long, to the perimeter of the square. Total, P=40x+8∙3x= 64x .

b) .

Let's move all horizontal "steps" and vertical upper segments to the outer area. The perimeter of the resulting rectangle is P=2(7y+4y)=22y. To find the perimeter of the original figure, you need to add to the perimeter of the rectangle the sum of the lengths of four segments, each with a length of y: P=22y+4∙y= 26y .

D) .

Move all the horizontal lines from the inner area to the outer area and move the two vertical outer lines in the left and right corners, respectively, z to the left and right. As a result, we get a large rectangle, the perimeter of which is P=2(11z+3z)=28z.

The perimeter of the original figure is equal to the sum of the perimeter of the large rectangle and the lengths of six segments in z: P=28z+6∙z= 34z .

b) .

The solution is completely similar to the solution of the previous example. After transforming the figure, we find the perimeter of the large rectangle:

P=2(5z+3z)=16z. To the perimeter of the rectangle we add the sum of the lengths of the remaining six segments, each of which is equal to z: P=16z+6∙z= 22z .

Rectangle - P = 2*a + 2*b = 2*3 + 2*6 = 6 + 12 = 18. In this problem, the perimeter coincided in value with the area of the figure.

Square Problem: find the perimeter of a square if its area is 9. Solution: using the square area formula S = a ^ 2, from here find the length of the side a = 3. The perimeter is equal to the sum of the lengths of all sides, therefore, P = 4 * a = 4 * 3 = 12.

Triangle Task: given an arbitrary ABC, the area of \u200b\u200bwhich is equal to 14. Find the perimeter of the triangle if the line drawn from the vertex B divides the base of the triangle into segments of length 3 and 4 cm. S = ½*AC*BE. The perimeter is equal to the sum of the lengths of all sides. Find the length of side AC by adding the lengths AE and EC, AC = 3 + 4 = 7. Find the height of the triangle BE = S*2/AC = 14*2/7 = 4. Consider right triangle A.B.E. Knowing AE and BE, you can find the hypotenuse using the Pythagorean formula AB^2 = AE^2 + BE^2, AB = √(3^2 + 4^2) = √25 = 5. Consider right triangle BEC. According to the Pythagorean formula BC^2 = BE^2 + EC^2, BC = √(4^2 + 4^2) = 4*√2. Now the lengths of all sides of the triangle. Find the perimeter from their sum P = AB + BC + AC = 5 + 4*√2 + 7 = 12 + 4*√2 = 4*(3+√2).

CircleProblem: it is known that the area of a circle is 16*π, find its perimeter. Solution: write down the formula for the area of a circle S = π*r^2. Find the radius of the circle r = √(S/π) = √16 = 4. According to the formula, the perimeter is P = 2*π*r = 2*π*4 = 8*π. If we accept that π = 3.14, then P = 8*3.14 = 25.12.

Sources:

area equals perimeter

All of us once in school begin to study the perimeter of a rectangle. So let's remember how to calculate it and what is the perimeter in general?

The word "perimeter" comes from two Greek words: "peri", which means "around", "about" and "metron", which means "to measure", "to measure". Those. perimeter, translated from Greek means "measurement around."

Instruction

The second definition will sound like this: the perimeter of a rectangle is twice the sum of its length and width.

Related videos

Helpful advice

The area of a rectangle is the product of its length times its width. Pemeter is the sum of all sides.

Sources:

A circle is a geometric figure formed from a set of points that are far from the center. circles on the equal distance. Based on the known circles data, there are 2 formulas arising from each other for determining its area.

You will need

The value of the constant π (equal to 3.14);
The size of the diameter/radius of a circle.

Instruction

Related videos

A square is a beautiful and simple flat geometric figure. This is a rectangle with equal sides. How to find perimeter square if the length of its side is known?

Instruction

First of all, remember that perimeter is nothing more than the sum of a geometric figure. Considered by us four sides. Moreover, by , all these sides are equal between .
From these premises, it is easy to find perimeter a square – perimeter square side length square multiplied by four:
P \u003d 4a, where a is the length of the side square.

Related videos

Tip 6: How to find the area of a triangle and a rectangle

Triangle and rectangle are the two simplest flat geometric figures in Euclidean geometry. Within the perimeters formed by the sides of these polygons, there is a certain section of the plane, the area of which can be determined in many ways. The choice of method in each particular case will depend on the known parameters of the figures.

Instruction

Use one of the trigonometric formulas to find the area of a triangle if you know the values of one or more angles in . For example, with a known value of the angle (α) and the lengths of the sides that make it up (B and C), the area (S) can be obtained by the formula S \u003d B * C * sin (α) / 2. And with the values of all angles (α, β and γ) and the length of one side in addition (A), you can use the formula S \u003d A² * sin (β) * sin (γ) / (2 * sin (α)). If, in addition to all angles, (R) of the circumscribed circle is known, then use the formula S=2*R²*sin(α)*sin(β)*sin(γ).

If the angles are not known, then to find the area of a triangle, you can use without trigonometric functions. For example, if (H) drawn from a side that also knows (A), then use the formula S \u003d A * H / 2. And if the lengths of each of the sides (A, B and C) are given, then first find the semi-perimeter p \u003d (A + B + C) / 2, and then calculate the area of \u200b\u200bthe triangle using the formula S \u003d √ (p * (p-A) * (p-B) * (p-C)). If, in addition to (A, B and C), the radius (R) of the circumscribed circle is known, then use the formula S \u003d A * B * C / (4 * R).

To find the area of a rectangle, you can also use trigonometric functions- for example, if the length of its diagonal (C) and the value of the angle that it has on one of the sides (α) are known. In this case, use the formula S=С²*sin(α)*cos(α). And if the lengths of the diagonals (C) and the angle they make up (α) are known, then use the formula S \u003d C² * sin (α) / 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? (Not.)
- 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(Annex 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).

A rectangle (or parallelogram) ABCD, then it has the following properties: the parallel sides are pairwise equal (see). AB = SD and AC = VD. Knowing the ratio of the sides in this figure, we can derive rectangle(and parallelogram): P \u003d AB + SD + AC + VD. Let some sides be equal to the number a, the other to the number b, then P \u003d a + a + b + b \u003d 2 * a \u003d 2 * b \u003d 2 * (a + c). Example 1. In ABCD, the sides are equal to AB = CD = 7 cm and AC = VD = 3 cm. Find the perimeter of such a rectangle. Solution: P \u003d 2 * (a + c). P \u003d 2 * (7 +3) \u003d 20 cm.

When solving problems for the sum of the lengths of the sides with a figure called a square or a rhombus, a slightly modified perimeter formula should be used. A square and a rhombus are shapes that have the same four sides. Based on the definition of the perimeter, P \u003d AB + SD + AC + VD and assuming lengths with the letter a, then P \u003d a + a + a + a \u003d 4 * a. Example 2. A rhombus of side 2 cm. Find its perimeter. Solution: 4*2 cm = 8 cm.

If the given quadrilateral is a trapezoid, then in this case you just need to add the lengths of its four sides. P \u003d AB + SD + AC + VD. Example 3. Find ABCD if its sides are equal: AB = 1 cm, SD = 3 cm, AC = 4 cm, ID = 2 cm. Solution: P = AB + SD + AC + ID = 1 cm + 3 cm + 4 cm + 2 cm = 10 cm. It may happen that it turns out to be equilateral (its two sides are equal), then its perimeter can be reduced to the formula: P \u003d AB + SD + AC + VD \u003d a + b + a + c \u003d 2*a + b + s. Example 4. Find the perimeter of an isosceles if its side faces are 4 cm, and the bases are 2 cm and 6 cm. Solution: P \u003d 2 * a + b + c \u003d 2 * 4 cm + 2 cm + 6 cm \u003d 16 cm.

Related videos

Helpful advice

Nobody bothers to find the perimeter of a quadrilateral (and any other figure) as the sum of the lengths of the sides, without using the derived formulas. They are given for convenience and ease of calculation. The solution method is not a mistake, the correct answer and knowledge of mathematical terminology are important.

Sources: