In the following test tasks you need to find the perimeter of the figure shown in the figure.
You can find the perimeter of a figure different ways. You can transform the original shape so 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, you cannot rely only on the drawing, but find the lengths of the segments based on the data of the problem.
I would like to warn you: in one of the tasks, among the proposed answer options, I did not find the one that worked 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 with a length of x, two with a length of 2x.
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 large rectangle, move one of the 4x length segments to the lower 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 three segments remaining inside P=2(10x+8x)+6x+4x+2x= 48x .
E) .
By transferring the inner sides of the 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 the 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 of length y: P=22y+4∙y= 26y .
D) .
Let's move all the horizontal lines from the inner area to the outer one and move the two vertical outer lines in the left and right corners, respectively, z to the left and to the right. As a result, we get a large rectangle whose perimeter 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 along 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 .
Geometry, if I’m not mistaken, in my time was studied from the fifth grade and perimeter was and is one of the key concepts. So, perimeter is the sum of the lengths of all sides (denoted by the Latin letter P). In general, this term is interpreted differently, for example,
- total length of the figure's border,
- the length of all its sides,
- the sum of the lengths of its faces,
- the length of the line limiting the figure,
- the sum of all the lengths of the sides of a polygon
Different figures have their own formulas for determining the perimeter. To understand the meaning, I propose to independently derive a few simple formulas:
- for a square,
- for a rectangle,
- for a parallelogram,
- for cube,
- for parallelepiped
Perimeter of a square
For example, let's take the simplest thing - the perimeter of a square.
All sides of the square are equal. Let one side be called "a" (as are the other three), then
P = a + a + a + a
or a more compact notation
Perimeter of a rectangle
Let's complicate the problem and take a rectangle. In this case, it is no longer possible to say that all sides are equal, so let the lengths of the sides of the rectangle be equal to a and b.
Then the formula will look like this:
P = a + b + a + b
Perimeter of a parallelogram
A similar situation will occur with a parallelogram (see the perimeter of the rectangle)
Cube perimeter
What to do if we are dealing with a three-dimensional figure? For example, let's take a cube. The cube has 12 sides and they are all equal. Accordingly, the perimeter of the cube can be calculated as follows:
Parallelepiped perimeter
Well, to secure the material, let’s calculate the perimeter of the parallelepiped. This requires some thought. Let's do this together. As we know, a rectangular parallelepiped is a figure whose sides are rectangles. Each parallelepiped has two bases. Let's take one of the bases and look at its sides - they have lengths a and b. Accordingly, the perimeter of the base is P = 2a + 2b. Then the perimeter of the two bases is
(2a + 2b) * 2 = 4a + 4b
But we also have a “c” side. This means that the formula for calculating the perimeter of a parallelepiped will be as follows:
P = 4a + 4b + 4c
As you can see from the examples above, all you need to do to determine the perimeter of a shape is to find the length of each side and then add them up.
In conclusion, I would like to note that not every figure has a perimeter. Eg, The ball has no perimeter.
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 summer cottage measuring 20 m by 30 m was purchased. Determine the area summer cottage, write your 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.
PERIMETER PERIMETER (from the Greek perimetreo - I measure), the length of a closed contour, for example the sum of the lengths of all sides of a polygon.
Modern encyclopedia. 2000 .
Synonyms:See what "PERIMETER" is in other dictionaries:
Perimeter... Spelling dictionary-reference book
Perimeter 2: New Earth Developer K D Lab Publisher 1C Release date... Wikipedia
- (Greek, from peri around, and metreo I measure). The sum of the sides of rectilinear figures. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. PERIMETER in Greek, from peri, around, and metreo, measuring. Circle of a polygon. Explanation... ... Dictionary of foreign words of the Russian language
Perimeter- the border of a protected area, equipped with physical barriers and checkpoints. Source … Dictionary-reference book of terms of normative and technical documentation
perimeter- a, m. perimètre m., German. Perimeter lat. perimetros circumference gr. peri about + metreo I measure. 1. mat. Sum of lengths of all sides geometric figure. BAS 1. || In rich houses at that time, sanitary ware from St. Laurence was installed. On the… … Historical Dictionary of Gallicisms of the Russian Language
PERIMETER, the length of the closed contour of a flat figure. The perimeter of a circle is the length of its CIRCLE. The perimeter of a polygon is equal to the sum of its sides... Scientific and technical encyclopedic dictionary
PERIMETER, perimeter, male. (Greek perimetron circle) (mat.). The sum of the lengths of all sides of a flat figure. Perimeter of a triangle. Perimeter of a polygon. Dictionary Ushakova. D.N. Ushakov. 1935 1940 ... Ushakov's Explanatory Dictionary
Length, border Dictionary of Russian synonyms. perimeter noun, number of synonyms: 2 border (39) length ... Synonym dictionary
- (from the Greek perimetreo I measure around) the length of a closed contour, for example. the sum of the lengths of all sides of a polygon... Big Encyclopedic Dictionary
PERIMETER, huh, husband. In mathematics: the boundary of a plane figure, as well as the length of this boundary. | adj. perimetric, oh, oh. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary
Books
- PERIMETER – Stuck Claw, Sergey Kochetkov. A former submariner, owing a large sum to a crime boss, puts his family at risk. To protect her, he embarks on a desperate adventure on the verge of life and treason. To him…