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инфоурок

школа

видеоуроки

геометрия

00:00:00

the concept of the area of a polygon

00:00:04

we know the concept of area from everyday

00:00:07

experience, everyone understands the meaning of the words get a

00:00:10

room

00:00:11

is equal to 16 square meters the area of a

00:00:15

garden plot is 800 com and so on in

00:00:19

this lesson we will look at the question of the

00:00:22

areas of a polygon we can say

00:00:26

that the area of a polygon

00:00:28

is the size of that part of the plane that

00:00:31

the polygon occupies measurement of

00:00:35

areas is carried out using the selected

00:00:38

unit of measurement,

00:00:40

similar to measuring the lengths of segments, the

00:00:44

unit of measurement of areas is taken to be a

00:00:47

square whose side is equal to the unit of

00:00:51

measurement of segments,

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so if a centimeter is taken as the unit of measurement of segments,

00:00:57

then a

00:01:01

square with a side of 1

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centimeter is taken as the unit of measurement of areas; such a square is called

00:01:08

square centimeters and is designated

00:01:12

This is how a square meter is defined in a similar way, a

00:01:16

square millimeter, and

00:01:19

so on, with the chosen unit of

00:01:23

area measurement, the

00:01:24

area of each polygon

00:01:26

is expressed as a positive number;

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this number shows how many times the unit of

00:01:33

measurement of its part

00:01:36

fits into a given polygon.

00:01:40

Consider examples. The figure shows a

00:01:43

rectangle in which a square

00:01:46

centimeter

00:01:47

fits exactly six times. this

00:01:51

means that the area of the rectangle

00:01:52

is equal to 6 square centimeters in the

00:01:58

trapezoid abcd the

00:02:00

square centimeter shown in the figure

00:02:03

is laid out twice and a

00:02:07

part of the trapezoid remains, a triangle from c to z

00:02:12

in which the square centimeter does not

00:02:14

fit entirely, to measure the

00:02:17

area of this triangle you need to

00:02:19

use a fraction of a square centimeter,

00:02:23

for example,

00:02:26

it is a square millimeter zero point one one

00:02:30

hundredth of a square centimeter this

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is shown in this figure where a

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square centimeter

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is divided into 100 square millimeters

00:02:41

this figure and also this figure

00:02:44

for greater clarity are

00:02:46

given on an enlarged scale in this

00:02:50

figure it can be seen that a square millimeter

00:02:53

fits into the triangle cdm 14 times and

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part remains this triangle in

00:03:02

which a square millimeter does not

00:03:04

fit entirely,

00:03:06

therefore we can say that the area of the

00:03:09

trapezoid abcd is approximately equal to

00:03:13

2.14 square centimeters, the remaining

00:03:18

part of the triangle cdm can be measured

00:03:21

using a smaller

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fraction of a square centimeter and a

00:03:26

more accurate value for the area of the trapezoid can be obtained; the

00:03:30

described measurement process can be

00:03:33

continued further, however in practice, it is

00:03:36

inconvenient,

00:03:39

usually they measure only some of the

00:03:41

segments associated with the polygon and then

00:03:44

calculate the area using certain

00:03:47

formulas; the

00:03:48

derivation of these formulas is based on the properties of

00:03:51

areas,

00:03:52

which we will now consider; first

00:03:56

of all, we note that if two polygons

00:03:59

are equal, then the units of measurement for areas and

00:04:03

parts

00:04:04

fit into such polygons are

00:04:06

the same number 1 that is,

00:04:09

the following properties hold:

00:04:11

equal polygons have equal

00:04:15

areas, then let the polygon be

00:04:19

composed of several polygons

00:04:22

so that the internal areas of

00:04:25

any two of these polygons do not

00:04:29

have common points, as shown in this

00:04:32

figure, obviously the

00:04:35

size of the part of the plane occupied by the entire

00:04:38

polygon is the sum of the sizes of

00:04:42

those parts of the plane which occupy the

00:04:46

constituent polygons and so

00:04:50

properties 2

00:04:51

if a polygon is made up of

00:04:54

several polygons then its

00:04:56

area is equal to the sum of the areas of these

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polygons properties 12

00:05:03

calls the main properties of areas let us

00:05:07

recall that the

00:05:10

lengths of segments have similar properties along with these

00:05:14

properties we will need one more

00:05:17

property of

00:05:18

areas properties 3 the area of a square

00:05:22

equal to the square of its side, a

00:05:25

short formulation of this property

00:05:28

should be understood as follows: if the side of the

00:05:31

square

00:05:32

with the chosen unit of measurement of segments

00:05:35

is expressed by a number, and then the area of this

00:05:38

square

00:05:39

is expressed by a number, and the square

00:05:43

in the figure shows a square

00:05:47

whose side is equal to 2 point one

00:05:49

centimeter, it consists of four

00:05:53

square centimeters and 41 square

00:05:57

millimeter,

00:05:58

so the area of a square is

00:06:03

4.41 square centimeters and

00:06:06

what is equal to the square of its side

00:06:10

4.41 is equal to two point one

00:06:14

squared, the proof of this

00:06:17

property is given in the next lesson

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