Today we will talk about how to calculate polygon perimeter. But first, let's talk about the variety of figures. Look at the drawing. 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.
In the following test tasks, you need to find the perimeter of the figure shown in the figure.
There are many ways to find the perimeter of a shape. 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) .
Moving 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 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 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 .
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:Perimeter ... Spelling Dictionary
Perimeter 2: New Earth Developer K D Lab Publisher 1C Release date ... Wikipedia
- (Greek, from peri around, and metreo I measure). Sum of sides of rectilinear figures. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. Greek PERIMETER, from peri, around, and metreo, I measure. Polygon circle. Explanation… … Dictionary of foreign words of the Russian language
Perimeter- the border of the protected area, equipped with physical barriers and checkpoints. Source … Dictionary-reference book of terms of normative and technical documentation
perimeter- a, m. perimètre m., germ. Perimeter lat. perimetros circumferencegr. peri about + metreo I measure. 1. mat. The sum of the lengths of all sides of a geometric figure. BAS 1. || In rich houses at that time, San Laurents plumbing was installed. On the… … Historical Dictionary of Gallicisms of the Russian Language
PERIMETER, the length of a closed contour of a flat figure. The perimeter of a circle is the length of its CIRCLE. The perimeter of a polygon is the sum of its sides... Scientific and technical encyclopedic dictionary
PERIMETER, perimeter, male. (Greek perimetron circumference) (mat.). The sum of the lengths of all sides of a plane figure. The perimeter of a triangle. The perimeter of the polygon. Explanatory Dictionary of Ushakov. D.N. Ushakov. 1935 1940 ... Explanatory Dictionary of Ushakov
Length, border Dictionary of Russian synonyms. perimeter n., number of synonyms: 2 border (39) length ... Synonym dictionary
- (from the Greek perimetreo I measure around) the length of a closed circuit, for example. the sum of the lengths of all sides of a polygon... Big Encyclopedic Dictionary
PERIMETER, a, husband. In mathematics: the boundary of a plane figure, as well as the length of this boundary. | adj. perimetric, oh, oh. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov
We do not use many formulas from the school mathematics course in everyday life. However, there are equations that are used, if not on a regular basis, then from time to time. One of these formulas is the calculation of the perimeter of a figure.
The perimeter is the total length of all sides of a geometric figure. For its designation, the letter of the Latin alphabet "R" is used. Simply put, to find the perimeter, you need to measure the lengths of all sides of a geometric figure and add the resulting values. The length is calculated with a conventional measuring instrument, such as a ruler, tape measure, centimeter tape, and so on.
The units of measurement are respectively centimeters, meters, millimeters and other measures of length. The length of a side of a polygon is calculated by applying a measuring device from one vertex to another. The beginning of the division scale of the device must coincide with one of the vertices. The second numeric value that the other vertex hits is the length of the side of the polygon. In the same way, it is necessary to measure all the lengths of the sides of the figure and add the resulting values. The unit of perimeter is the same unit used to measure the side of a figure.
A rectangle should be called a geometric figure, which consists of four sides of different lengths and three corners of which are straight. When constructing such a figure on a plane, it turns out that its sides will be equal in pairs, but not all are equal to each other. What is the perimeter of a rectangle? It is also the total length of all figure lengths. But since two sides of a rectangle have the same value, in calculating the perimeter, you can add the lengths of two adjacent sides twice. The unit of measurement for the perimeter of a rectangle is also the generally accepted units of measurement.
A triangle should be called a geometric figure that has three angles (both of different values and the same) and consists of segments formed from the points of intersection of the rays that form the angles. A triangle has three sides and three angles. Two of the three sides can be equal in it. Such a triangle should be considered isosceles. There are such figures in which all three sides are equal to each other. It is customary to call such triangles equilateral.
What is the perimeter of a triangle? Its calculation can be carried out by analogy with the perimeter of a quadrilateral. The perimeter of a triangle is equal to the total length of the lengths of its sides. Calculating the perimeter of a triangle in which two sides are equal - isosceles - is simplified by multiplying one length of equal sides by two. To the value obtained, you must add the value of the length of the third side. Calculating the perimeter of a triangle with equal sides can be reduced to a simple calculation of the product of one length of a side of a triangle by three.
The calculation of the perimeter in everyday life is used in many areas, but most often when performing construction, geodetic, topographic, architectural, planning work. But the listed scope of perimeter calculation, of course, is not limited.
For example, when performing geodetic and topographic works, it is very often necessary to calculate the perimeter of the boundaries of a certain area. But in practice, the plots rarely have the correct shape. Therefore, the calculation of the length of the perimeter occurs according to the formula for calculating the sum of the lengths of all sides of the section.
The need to calculate the perimeter of the site is very often due to the fact that you need to know how much material is required to install fences. Even a simple personal plot needs to measure the perimeter in order to competently enclose it with a fence.
To calculate the perimeter on the ground, it is impossible to use a simple student ruler. Therefore, specialists use special devices. Of course, the simplest and most affordable option is to measure the length of the site boundary in steps. The step size of an adult is approximately one meter. Sometimes one meter and twenty centimeters. But this method is very inaccurate and gives a large measurement error. It is suitable if there is no need to accurately calculate the length of the border, but there is a need to simply estimate the approximate length.
For a more accurate calculation of the length of the sides of the site and, accordingly, the perimeter, there are special devices. First of all, you can use a special metal tape measure or a regular wire.
There are also special measuring devices such as rangefinders. Devices are optical, laser, light, ultrasonic. It should be remembered that the farther the rangefinder is able to measure the distance, the higher its error. Such devices are used in geodetic and topographic surveys.
Geometry, if I'm not mistaken, in my time was studied from the fifth grade and the 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 in different ways, for example,
Different shapes have their own formulas for determining the perimeter. To understand the meaning itself, I propose to independently deduce a few simple formulas:
For example, let's take the simplest - the perimeter of a square.
All sides of a square are equal. Let one side be called "a" (as well as the other three), then
P = a + a + a + a
or more compact notation
Let's complicate the task 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
A similar situation will be with a parallelogram (see the perimeter of the rectangle)
What to do if we are dealing with a three-dimensional figure? For example, take a cube. A cube has 12 sides and they are all equal. Accordingly, the perimeter of a cube can be calculated as follows:
Well, to fix the material, we calculate the perimeter of the parallelepiped. Here it is necessary to think a little. Let's do it together. As we know, a cuboid 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. So the formula for calculating the perimeter of a parallelepiped will look like this:
P = 4a + 4b + 4c
As you can see from the examples above, all that needs to be done to determine the perimeter of a shape is to find the length of each of the sides, and then add them up.
In conclusion, I would like to note that not every figure has a perimeter. Eg, A sphere has no perimeter.
