In this lesson, students are given the opportunity to get acquainted with another unit of area measurement, square decimeter, learn how to convert square decimeters to square centimeters, and also practice in performing various tasks on comparing values \u200b\u200band solving problems on the lesson.

Read the topic of the lesson: "The unit of area is a square decimeter." In the lesson we will get acquainted with another unit of area, square decimeter, learn how to convert square decimeters to square centimeters and compare values.

Draw a rectangle with sides of 5 cm and 3 cm and mark its vertices with letters (Fig. 1).

Fig. 1. Illustration of the problem

Find the area of \u200b\u200bthe rectangle. To find the area, it is necessary to multiply the length by the width of the rectangle.

We write the solution.

5 * 3 \u003d 15 (cm 2)

Answer: the area of \u200b\u200bthe rectangle is 15 cm 2.

We calculated the area of \u200b\u200bthis rectangle in square centimeters, but, sometimes, depending on the problem being solved, the units of area can be different: more or less.

The area of \u200b\u200bthe square, the side of which is 1 dm, is the unit of area, square decimeter(fig. 2) .

Fig. 2. Square decimeter

The words “square decimeter” with numbers are written like this:

5 dm 2, 17 dm 2

Set the ratio between the square decimeter and the square centimeter.

Since a square with a side of 1 dm can be divided into 10 strips, each of which has 10 cm 2, then only ten dozen, or one hundred square centimeters in a square decimeter (Fig. 3).

Fig. 3. One hundred square centimeters

Remember.

1 dm 2 \u003d 100 cm 2

Express these values \u200b\u200bin square centimeters.

5 dm 2 \u003d ... cm 2

8 dm 2 \u003d ... cm 2

3 dm 2 \u003d ... cm 2

We reason so. We know that in one square decimeter one hundred square centimeters, it means that in five square decimeters five hundred square centimeters.

Check yourself.

5 dm 2 \u003d 500 cm 2

8 dm 2 \u003d 800 cm 2

3 dm 2 \u003d 300 cm 2

Express these values \u200b\u200bin square decimeters.

400 cm 2 \u003d ... dm 2

200 cm 2 \u003d ... dm 2

600 cm 2 \u003d ... dm 2

We explain the decision. One hundred square centimeters make up one square decimeter, which means that there are four square decimeters in the number of 400 cm 2.

Check yourself.

400 cm 2 \u003d 4 dm 2

200 cm 2 \u003d 2 dm 2

600 cm 2 \u003d 6 dm 2

Perform actions.

23 cm 2 + 14 cm 2 \u003d ... cm 2

84 dm 2 - 30 dm 2 \u003d ... dm 2

8 dm 2 + 42 dm 2 \u003d ... dm 2

36 cm 2 - 6 cm 2 \u003d ... cm 2

Consider the first expression.

23 cm 2 + 14 cm 2 \u003d ... cm 2

We add the numerical values: 23 + 14 \u003d 37 and assign the name: cm 2. We continue to reason similarly.

Check yourself.

23 cm 2 + 14 cm 2 \u003d 37 cm 2

84 dm 2 - 30 dm 2 \u003d 54 dm 2

8 dm 2 + 42 dm 2 \u003d 50 dm 2

36 cm 2 - 6 cm 2 \u003d 30 cm 2

Read and solve the problem.

The height of the rectangular mirror is 10 dm, and the width is 5 dm. What is the area of \u200b\u200bthe mirror (Fig. 4)?

Fig. 4. Illustration to the task

To find out the area of \u200b\u200ba rectangle, you need to multiply the length by the width. Note that both values \u200b\u200bare expressed in decimeters, which means that the area name will be dm 2.

We write the solution.

5 * 10 \u003d 50 (dm 2)

Answer: the area of \u200b\u200bthe mirror is 50 dm 2.

Compare values.

20 cm 2 ... 1 dm 2

6 cm 2 ... 6 dm 2

95 cm 2 ... 9 dm

It is important to remember: in order to compare values, they must have the same name.

Consider the first line.

20 cm 2 ... 1 dm 2

We convert square decimeter to square centimeter. Remember that in one square decimeter one hundred square centimeters.

20 cm 2 ... 1 dm 2

20 cm 2 ... 100 cm 2

20 cm 2< 100 см 2

Consider the second line.

6 cm 2 ... 6 dm 2

We know that square decimeters are larger than square centimeters, and the numbers with these names are the same, which means we put the sign “<».

6 cm 2< 6 дм 2

Consider the third line.

95cm 2 ... 9 dm

Note that the units of area are recorded on the left, and the linear units are written on the right. Such values \u200b\u200bcannot be compared (Fig. 5).

Fig. 5. Different sizes

Today at the lesson we got acquainted with another unit of area, square decimeter, learned to convert square decimeters to square centimeters and compare values.

On this our lesson is over.

Bibliography

M.I. Moreau, MA Bantova et al. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
M.I. Moreau, MA Bantova et al. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
M.I. Moro. Lessons of mathematics: Methodical recommendations for the teacher. 3 class. - M .: Education, 2012.
Regulatory document. Monitoring and evaluation of learning outcomes. - M .: "Enlightenment", 2011.
"School of Russia": Programs for elementary school. - M .: "Enlightenment", 2011.
S.I. Volkov. Mathematics: Test work. 3 class. - M .: Education, 2012.
V.N. Rudnitskaya. Tests - M .: “Exam”, 2012.

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Homework

1. The length of the rectangle - 7 dm, width - 3 dm. What is the area of \u200b\u200ba rectangle?

2. Express these values \u200b\u200bin square centimeters.

2 dm 2 \u003d ... cm 2

4 dm 2 \u003d ... cm 2

6 dm 2 \u003d ... cm 2

8 dm 2 \u003d ... cm 2

9 dm 2 \u003d ... cm 2

3. Express these values \u200b\u200bin square decimeters.

100 cm 2 \u003d ... dm 2

300 cm 2 \u003d ... dm 2

500 cm 2 \u003d ... dm 2

700 cm 2 \u003d ... dm 2

900 cm 2 \u003d ... dm 2

4. Compare the values.

30 cm 2 ... 1 dm 2

7 cm 2 ... 7 dm 2

81 cm 2 ... 81 dm

5. Make a task for your fellow lesson.

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1 square decimeter [dm²] \u003d 100 square centimeter [cm²]

Baseline

Converted value

square meter square kilometer square hectometer square decameter square decimeter square centimeter square millimeter square micrometer square nanometer hectare ar barn square mile square meter mile (United States, surveying.) square yard square foot² square. ft (US, geodesic.) square inch circular inch township section acre acre (US, geodesic) ores square chain square rod² (US, geodesic) square square square square. thousandth circular mil homestead sabin arpan quaeda square castilian elbow varas conuqueras cuad electron cross section tithe (breech) tithe economic round square verst square arshin square foot square sazhen square inch (russian) square line plank square

More about the area

General information

Area is the value of a geometric figure in two-dimensional space. It is used in mathematics, medicine, engineering, and other sciences, for example, in calculating the cross section of cells, atoms, or tubes, such as blood vessels or water pipes. In geography, an area is used to compare the size of cities, lakes, countries, and other geographic features. The area is also used in calculating population density. Population density is defined as the number of people per unit area.

Units

Square meters

The area is measured in the SI system in square meters. One square meter is a square with a side of one meter.

Unit square

A unit square is a square with sides of one unit. The area of \u200b\u200ba unit square is also equal to one. In the rectangular coordinate system, this square is located in the coordinates (0,0), (0,1), (1,0) and (1,1). On the complex plane, the coordinates are 0, 1, i and i+1 where i - imaginary number.

Ar

Ap or hectare, as a measure of area, is used in the CIS countries, Indonesia and some other European countries to measure small urban sites such as parks, when the hectare is too large. One AP equals 100 square meters. In some countries, this unit is called otherwise.

Hectare

In hectares measure real estate, especially land. One hectare is 10,000 square meters. It has been used since the French Revolution, and is used in the European Union and some other regions. Like ar, in some countries hectares are called differently.

Acre

In North America and Burma, area is measured in acres. Hectares are not used there. One acre is equal to 4046.86 square meters. Initially, an acre was defined as the area that a peasant could plow in one day with a team of two oxen.

Barn

Barns are used in nuclear physics to measure the cross section of atoms. One barn is 10² square meters. Barn is not a unit in the SI system, but accepted for use in this system. One barn is approximately equal to the cross-sectional area of \u200b\u200bthe uranium nucleus, which physicists jokingly called "as huge as a barn." The barn in English "barn" (pronounced barn) and from the jokes of physicists, this word became the name of the unit area. This unit originated during the Second World War, and the scientists liked it, because its name could be used as a code in correspondence and telephone conversations within the framework of the Manhattan project.

Area calculation

The area of \u200b\u200bthe simplest geometric figures is found by comparing them with the square of a known area. This is convenient because the square of the square is easy to calculate. Some formulas for calculating the area of \u200b\u200bgeometric figures given below are obtained in this way. Also, to calculate the area, especially a polygon, the figure is divided into triangles, the area of \u200b\u200beach triangle is calculated by the formula, and then added. The area of \u200b\u200bmore complex shapes is calculated using mathematical analysis.

Formula for calculating the area

Square: side squared.
Rectangle: product of the parties.
Triangle (known side and height): the product of the side and the height (the distance from this side to the edge), divided in half. Formula: A \u003d ½ahwhere A - square, a - side and h - height.
Triangle (two sides are known and the angle between them): the product of the sides and the sine of the angle between them, divided in half. Formula: A \u003d ½ab sin (α) where A - square, a and b - sides, and α - the angle between them.
Equilateral triangle: the side squared divided by 4 and multiplied by the square root of three.
Parallelogram: the product of the side and height measured from this side to the opposite.
Trapeze: the sum of two parallel sides multiplied by the height and divided by two. Height is measured between these two sides.
A circle: product of the square of the radius and π.
Ellipse: the product of semi-axes and π.

Surface area calculation

Find the surface area of \u200b\u200bsimple volumetric shapes, such as prisms, by scanning this shape on a plane. The development of the ball can not be obtained in this way. The surface area of \u200b\u200bthe ball is found using a formula, multiplying the square of the radius by 4π. From this formula it follows that the area of \u200b\u200ba circle is four times smaller than the surface area of \u200b\u200ba ball with the same radius.

Surface areas of some astronomical objects: the Sun - 6.088 x 10¹² square kilometers; Earth - 5.1 x 10⁸; thus, the surface area of \u200b\u200bthe earth is about 12 times smaller than the surface area of \u200b\u200bthe sun. The surface area of \u200b\u200bthe moon is approximately 3.793 x 10⁷ square kilometers, which is about 13 times smaller than the surface area of \u200b\u200bthe earth.

Planimeter

The area can also be calculated using a special device - planimeter. There are several types of this device, for example polar and linear. Also, planimeters are analog and digital. In addition to other functions, you can enter a scale in digital planimeters, which makes it easier to measure objects on the map. The planimeter measures the distance traveled along the perimeter of the object being measured, as well as the direction. The distance covered by the planimeter parallel to its axis is not measured. These devices are used in medicine, biology, technology, and agriculture.

Area property theorem

According to the isoperimetric theorem, of all the figures with the same perimeter, the largest area of \u200b\u200bthe circle. If, on the contrary, to compare figures with the same area, then the circle has the smallest perimeter. The perimeter is the sum of the lengths of the sides of a geometric figure, or the line that denotes the boundaries of this figure.

Geographical objects with the largest area

Country: Russia, 17,098,242 square kilometers, including land and water area. The second and third largest countries in the country are Canada and China.

City: New York is the city with the largest area of \u200b\u200b8683 square kilometers. The second largest city is Tokyo, occupying 6993 square kilometers. The third is Chicago, with an area of \u200b\u200b5,498 square kilometers.

City Square: The largest square covering 1 square kilometer is located in the capital of Indonesia, Jakarta. This is Medan Merdeka Square. The second largest area of \u200b\u200b0.57 square kilometer is Praça duz Girazois in the city of Palmas, in Brazil. The third largest - Tiananmen Square in China, 0.44 square kilometers.

Lake: Geographers argue whether the Caspian Sea is a lake, but if so, this is the largest lake in the world with an area of \u200b\u200b371,000 square kilometers. The second largest lake is Lake Superior in North America. This is one of the lakes of the Great Lakes system; its area is 82,414 square kilometers. The third largest is Lake Victoria in Africa. It covers an area of \u200b\u200b69,485 square kilometers.

Choose a tape measure or measuring tape. Choose a tape measure or measuring tape with markings on it in centimeters (cm) or meters (m). This fixture will facilitate the calculation of square meters, as they were developed in the same measurement system.

If you managed to find a tape measure in feet or inches, measure the area using the available units of measure, and then go to the step that describes how to convert other units of measure into square meters.

Measure the length of the area you selected. A square meter is a unit of measurement for the area or size of a two-dimensional object such as a floor or field. Measure the length of one side from one corner to the other and record the result.

If the length is more than one meter, then count both meters and centimeters. For example, 2 meters 35 centimeters.
If the object that you measure is not a rectangle or a square, then read the third section of this article - "Measure the area of \u200b\u200bcomplex shapes."

If you cannot measure length at a time, do it in stages. Lay out the tape and mark where it ended (for example, 1 meter or 25 centimeters), then lay it again and start from the marked area. Repeat until you measure the full length. Then add all the measurements together.

Measure width. Use the same tape measure to measure the width of the object. The measurement must be started by placing the tape measure at an angle of 90º relative to the length of the object that you have already measured. That is, two lines of the square adjacent to each other. The resulting numbers also write on paper.

If the measured length is slightly less than one meter, then round to the nearest centimeter when taking measurements. For example, if the width is slightly more than 1 meter 8 centimeters, then just write down "1 m. 8 cm." and do not count millimeters.

Translate centimeters to meters. Usually, measurements cannot be made exactly in meters. You will get indicators in both meters and centimeters, for example, "2 meters 35 centimeters." 1 centimeter \u003d 0.01 meters, and therefore you can convert centimeters to meters if you move the comma 2 digits to the left. Here are some examples.

35cm \u003d 0.35m, so 2m 35cm \u003d 2m + 0.35m \u003d 2.35m
8cm \u003d 0.08m, so 1m 8cm \u003d 1.08m

Multiply the length by the width. Once you translate all measurements into meters, multiply the length by the width and get the area of \u200b\u200bthe object to be measured. If necessary, use the calculator. For example:

2.35mx 1.08m \u003d 2.538 square meters (m 2).

Round up. If you get a lot of digits after the decimal point, for example, 2.538 square meters, then round up, for example, 2.54 square meters. It is likely that you did not measure to the nearest millimeter, so the last figures will still not be accurate. In most cases, we round to the nearest centimeter (0.01m). If you need more accurate measurements, read this material.

Each time you multiply two numbers with the same units of measurement (for example, meters), the answer must be recorded in the same unit of measurement (m 2 or square meters).

1 square meter [m²] \u003d 100 square decimeter [dm²]

Baseline

Converted value

Ferromagnetic fluids

More about the area

General information

Units

Square meters

The area is measured in the SI system in square meters. One square meter is a square with a side of one meter.

Unit square

Ar

Hectare

Acre

Barn

Area calculation

Formula for calculating the area

Square: side squared.
Rectangle: product of the parties.
Triangle (known side and height): the product of the side and the height (the distance from this side to the edge), divided in half. Formula: A \u003d ½ahwhere A - square, a - side and h - height.
Triangle (two sides are known and the angle between them): the product of the sides and the sine of the angle between them, divided in half. Formula: A \u003d ½ab sin (α) where A - square, a and b - sides, and α - the angle between them.
Equilateral triangle: the side squared divided by 4 and multiplied by the square root of three.
Parallelogram: the product of the side and height measured from this side to the opposite.
Trapeze: the sum of two parallel sides multiplied by the height and divided by two. Height is measured between these two sides.
A circle: product of the square of the radius and π.
Ellipse: the product of semi-axes and π.

Surface area calculation

Planimeter

Area property theorem

Geographical objects with the largest area

Country: Russia, 17,098,242 square kilometers, including land and water area. The second and third largest countries in the country are Canada and China.