Definition.
Rectangle - This is a quadrilateral of which two opposite sides are equal to all four angle of the same.Rectangles differ in each other only by the ratio of the long side to the short, but all four corners have direct, that is, 90 degrees.
The long side of the rectangle is called length rectangle, and short - rectangle width.
The side of the rectangle is simultaneously its heights.
The main properties of the rectangle
Rectangle can be parallelogram, square or rhombus.
1. Opposite directions of the rectangle have the same length, that is, they are equal:
AB \u003d CD, BC \u003d AD
2. Opposite directions of the rectangle are parallel:
3. The adjacent directions of the rectangle are always perpendicular to:
AB ┴ BC, BC ┴ CD, CD ┴ AD, AD ┴ AB
4. All four rectangle corners are direct:
∠abc \u003d ∠bcd \u003d ∠cda \u003d ∠dab \u003d 90 °
5. The sum of the corners of the rectangle is 360 degrees:
∠abc + ∠bcd + ∠cda + ∠dab \u003d 360 °
6. The diagonal of the rectangle has the same length:
7. The sum of the squares of the diagonal of the rectangle is equal to the sum of the squares of the sides:
2D 2 \u003d 2A 2 + 2B 2
8. Each diagonal of the rectangle divides the rectangle into two identical figures, namely on rectangular triangles.
9. The diagonals of the rectangle intersect and at the intersection point are divided by half:
| AO \u003d BO \u003d CO \u003d DO \u003d | d. | ||
| 2 |
10. The intersection point of the diagonals is called the center of the rectangle and is also the center of the described circle
11. The diagonal of the rectangle is the diameter of the described circle
12. A circle can always be described around the rectangle, since the sum of opposite angles is 180 degrees:
∠abc \u003d ∠cda \u003d 180 ° ∠bcd \u003d ∠dab \u003d 180 °
13. In a rectangle, in which the length is not equal to the width, it is impossible to enter the circle, since the sums of opposite sides are not equal to each other (it is possible to enter the circle only in a special case of a rectangle - a square).
Side of the rectangle
Definition.
Length rectangle Call the length of a longer pair of his sides. Rectangle width Call the length of a shorter pair of his sides.Formulas for determining the lengths of the side of the rectangle
1. Formula of the side of the rectangle (the lengths and widths of the rectangle) through the diagonal and the other side:
a \u003d √ d 2 - B 2
b \u003d √ d 2 - A 2
2. Formula of the side of the rectangle (the lengths and widths of the rectangle) through the area and the other side:
| b \u003d D COS | β |
| 2 |
Diagonal rectangle
Definition.
Diagonal rectangle It is called any segment connecting two vertices of opposite corners of the rectangle.Formulas for determining the length of the diagonal of the rectangle
1. The formula for the diagonal of the rectangle through the two sides of the rectangle (through the Pythagore's theorem):
d \u003d √ a 2 + b 2
2. The formula of the diagonal of the rectangle through the area and any side:
4. The formula of the diagonal of the rectangle through the radius of the circle described:
d \u003d 2r.
5. Formula of the diagonal of the rectangle through the diameter of the described circle:
d \u003d D about
6. The formula of the diagonal of the rectangle through the sine of the angle adjacent to the diagonal, and the length of the side opposite to this corner:
8. The formula of the diagonal of the rectangle through the sine of a sharp corner between the diagonals and the area of \u200b\u200bthe rectangle
d \u003d √2s: sIN β.
Perimeter of rectangle
Definition.
Perimeter rectangle The sum of the lengths of all sides of the rectangle is called.Formulas for determining the length of the perimeter of the rectangle
1. Formula Perimeter of the rectangle through two sides of the rectangle:
P \u003d 2A + 2B
P \u003d 2 (A + B)
2. Formula Perimeter of the rectangle through the area and any side:
| P \u003d. | 2S + 2A 2 | = | 2S + 2B 2 |
| a. | b. |
3. Formula Perimeter of the rectangle through a diagonal and any side:
P \u003d 2 (a + √ d 2 - A 2) \u003d 2 (B + √ d 2 - B 2)
4. Formula of the perimeter of the rectangle through the radius of the circle described and any side:
P \u003d 2 (a + √4r 2 - a 2.) \u003d 2 (B + √4r 2 - b 2.)
5. Formula Perimeter of the rectangle through the diameter of the described circle and any side:
P \u003d 2 (A + √D O 2 - a 2.) \u003d 2 (B + √D O 2 - b 2.)
Square rectangle
Definition.
Square rectangle It is called space limited by the sides of the rectangle, that is, within the perimeter of the rectangle.Rectangle Square Definition Formulas
1. The formula of the rectangle area through two sides:
S \u003d A · B
2. The formula of the area of \u200b\u200bthe rectangle through the perimeter and any side:
5. The formula of the rectangle area through the radius of the circle described and any side:
S \u003d a √4r 2 - a 2. \u003d B √4r 2 - b 2.
6. The formula of the rectangle area through the diameter of the circle described and any side:
S \u003d a √d o 2 - a 2. \u003d B √D O 2 - b 2.
Circle described around a rectangle
Definition.
Circle described around a rectangle The circle passing through four vertices of the rectangle, the center of which lies at the intersection of the diagonals of the rectangle.The formulas for determining the circle radius described around the rectangle
1. The formula for the circle radius described around the rectangle through two sides:
Lesson and presentation on the topic: "Perimeter and Rectangle Square"
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What is a rectangle and square
Rectangle - This is a quadrilateer who has all the corners direct. So, the opposite directions are equal to each other.
Square - This is a rectangle that also is equal and side, and corners. It is called the right quadricle.
Quarterly, including rectangles and squares, are designated 4 letters - vertices. Latin letters are used to designate vertices: A, B, C, D...
Example. 
It is read like this: a four-trigger ABCD; Square EFGH.
What is the perimeter of the rectangle? Perimeter calculation formula
Perimeter of rectangle - This is the sum of the lengths of all sides of the rectangle or the sum of length and width multiplied by 2.Perimeter is indicated by the Latin letter P.. Since the perimeter is the length of all sides of the rectangle, it is perimeter written in units of length: mm, cm, m, dm, km.
For example, the perimeter of the AVD rectangle is indicated as P. ABCD, where A, B, C, D are the tops of the rectangle.
We write the perimeter formula of the ABCD quadrilateral:
P abcd \u003d ab + bc + cd + ad \u003d 2 * ab + 2 * bc \u003d 2 * (AB + BC)
Example.
ABCD rectangle is set with sides: AB \u003d Cd \u003d 5 cm and ad \u003d bc \u003d 3 cm.
Determine P ABCD.
Decision:
1. Draw an ABCD rectangle with source data. 
2. We write a formula for calculating the perimeter of this rectangle:
P. ABCD \u003d 2 * (AB + BC)
P. Abcd \u003d 2 * (5 cm + 3 cm) \u003d 2 * 8 cm \u003d 16 cm
Answer: p abcd \u003d 16 cm.
The formula for calculating the perimeter of the square
We have a formula for determining the perimeter of the rectangle.P. ABCD \u003d 2 * (AB + BC)
Apply it to determine the perimeter of the square. Considering that all sides of the square are equal, we get:
P. ABCD \u003d 4 * AB
Example.
ABCD square is set to a side equal to 6 cm. We define the perimeter of the square.
Decision.
1. Draw an ABCD square with source data.
2. Recall the calculation formula of the perimeter of the square:
P. ABCD \u003d 4 * AB
3. Substitute our data in the formula:
P. Abcd \u003d 4 * 6 cm \u003d 24 cm
Answer: p abcd \u003d 24 cm.
Tasks for finding the perimeter of the rectangle
1. Measure the width and length of rectangles. Determine their perimeter. 
2. Draw a rectangle ABCD with sides of 4 cm and 6 cm. Determine the perimeter of the rectangle.
3. Draw a SEOM square with a side of 5 cm. Determined the perimeter of the square.
Where is the calculation of the perimeter of the rectangle?
1. The land plot is set, it needs to be removed by the fence. What length will be a fence?
In this task, it is necessary to accurately calculate the perimeter of the site so as not to buy extra material for the construction of the fence.
2. Parents decided to make repairs in the children's room. It is necessary to know the perimeter of the room and its area to correctly calculate the number of wallpaper.
Determine the length and width of the room in which you live. Determine the perimeter of their room.
What is a rectangle area?
Area - This is the numerical characteristics of the figure. The area is measured by square units of length: cm 2, m 2, dm 2, etc. (centimeter in square, meter in square, decimeter in square, etc.)The calculations denotes the Latin letter S..
To determine the area of \u200b\u200bthe rectangle, it is necessary to multiply the length of the rectangle on its width. 
The area of \u200b\u200bthe rectangle is calculated by multiplying the length of the AK to the width of the CM. We write it as a formula.
S. AKMO \u003d AK * KM
Example.
What is the square of the AKMO rectangle, if its parties are 7 cm and 2 cm?
S. AKMO \u003d AK * KM \u003d 7 cm * 2 cm \u003d 14 cm 2.
Answer: 14 cm 2.
Formula for calculating square square
The square of the square can be determined by multiplying the side itself.Example. 
In this example, the square of the square is calculated multiplying the AB side to the BC width, but since they are equal, it turns out the multiplication of the AB side to AB.
S. AVCO \u003d AB * BC \u003d AB * AB
Example.
Determine the square of AKMO Square with a side of 8 cm.
S. AKMO \u003d AK * KM \u003d 8 cm * 8 cm \u003d 64 cm 2
Answer: 64 cm 2.
Tasks for finding the square of the rectangle and square
1. Rectangle with the sides of 20 mm and 60 mm. Calculated its area. Write an answer in square centimeters.2. A cottage plot was bought in size 20 m per 30 m. Determine the area of \u200b\u200bthe country area, the answer is written in square centimeters.
We have already met with the concept square Figure, learned one of the units of Square - square centimeter. In the lesson, we derive the rule how to calculate the area of \u200b\u200bthe rectangle.
We already know how to find the area of \u200b\u200bthe figures that are divided into square centimeters.
For example:
We can determine that the area of \u200b\u200bthe first figure is 8 cm 2, the area of \u200b\u200bthe second figure is 7 cm 2.
How to find a rectangle area, the length of the side of which is 3 cm and 4 cm?
To solve the problem, we break a rectangle for 4 strips of 3 cm 2 each.
Then the area of \u200b\u200bthe rectangle will be 3 * 4 \u003d 12 cm 2.
The same rectangle can be divided into 3 strips of 4 cm 2.
Then the area of \u200b\u200bthe rectangle will be 4 * 3 \u003d 12 cm 2.
In both cases to find the area of \u200b\u200bthe rectangle, the numbers expressing the lengths of the sides of the rectangle are multiplied.
We find the area of \u200b\u200beach rectangle.
Consider the Akmo Rectangle.
In one strip 6 cm 2, and such strips in this rectangle 2. Therefore, we can do the following:
The number 6 denotes the length of the rectangle, and 2 is the width of the rectangle. Thus, we changed the side of the rectangle in order to find the area of \u200b\u200bthe rectangle.
Consider the rectangle KDCO.
In the KDCO rectangle in one strip 2cm 2, and such strips 3. Therefore, we can perform action
Number 3 denotes the length of the rectangle, and 2 - the width of the rectangle. We changed them and recognized the area of \u200b\u200bthe rectangle.
We can conclude: to find the area of \u200b\u200bthe rectangle, you do not need to break the figure per square centimeters every time.
To calculate the area of \u200b\u200bthe rectangle, it is necessary to find it length and width (the lengths of the sides of the rectangle must be expressed in the same units of measurement), and then calculate the product of the numbers obtained (the area will be expressed in the respective units of Square)
Summarizing: the area of \u200b\u200bthe rectangle is equal to the product of its length and width.
Decide the task.
Calculated the area of \u200b\u200bthe rectangle, if the length of the rectangle is 9cm, and the width is 2 cm.
We argue like that. In this task, the length and width of the rectangle are known. Therefore, we act according to the rule: the area of \u200b\u200bthe rectangle is equal to the product of its length and width.
We write down the decision.
Answer: Rectangle area 18cm 2
What do you think, what else can be the length of the side of the rectangle with such an area?
You can talk like that. Since the area is a product of the lengths of the rectangle, so you need to recall the multiplication table. When multiplying what numbers is the answer 18?
That's right, with multiplication 6 and 3, too, it will turn out 18. So, a rectangle can be part of 6cm and 3 cm and its area will also be equal to 18 cm 2.
Decide the task.
The length of the rectangle is 8cm, and width 2cm. Find its area and perimeter.
We know the length and width of the rectangle. It is necessary to remember that in order to find the area it is necessary to find a product of its length and width, and to find the perimeter you need the sum of length and width multiplied by two.
We write down the decision.
Answer: The area of \u200b\u200bthe rectangle is 16 cm 2, and the perimeter of the rectangle is 20 cm.
Decide the task.
The length of the rectangle is 4cm, and the width is 3 cm. What is the triangle square? (see drawing)
To answer the question of the task, you first need to find a rectangle area. We know that for this you need to multiply the length to the width.
Look at the drawing. Did you notice the diagonal divided the rectangle into two equal triangles? Consequently, the area of \u200b\u200bone triangle is 2 times less than the area of \u200b\u200bthe rectangle. So, it is necessary to reduce 12 by 2 times.
Answer: The area of \u200b\u200bthe triangle is 6 cm 2.
Today, at the lesson, we met the rule, how to calculate the area of \u200b\u200bthe rectangle and learned to apply this rule when solving problems on finding a rectangle area.
1. M.I. Moro, M.A.Bantova and others. Mathematics: Tutorial. Grade 3: In 2 parts, part 1. M., "Enlightenment", 2012.
2. M.I. Moro, M.A.Bantova and others. Mathematics: Tutorial. Grade 3: In 2 parts, part 2. M., "Enlightenment", 2012.
3. M.I. Moro. Mathematics lessons: Methodical recommendations for the teacher. Grade 3. - M.: Enlightenment, 2012.
4. Regulatory document. Control and evaluation of learning outcomes. M., "Enlightenment", 2011.
5. School of Russia: Primary School Programs. - M.: "Enlightenment", 2011.
6. S.I. Volkov. Mathematics: test work. Grade 3. - M.: Enlightenment, 2012.
7. V.N. Rodnitskaya. Tests. M., "Exam", 2012 (127С.)
2. Publisher "Enlightenment" ()
1. The length of the rectangle is 7 cm, width 4 cm. Find the area of \u200b\u200bthe rectangle.
2. Square side 5 cm. Find the square area.
3. Inclinee possible options for rectangles whose area is 18 cm 2.
4. Make a task on the subject of the lesson for your comrades.
Periodically, we need to know the area and the volume of the room. These data may be needed when designing heating and ventilation, when purchasing building materials and in many other situations. Also periodically need to know the walls of the walls. All this data is calculated easily, but will pre-work a roulette - measure all the required dimensions. On how to calculate the area of \u200b\u200bthe room and the walls, the size of the room and will be discussed further.
Room Square in Square meters
- Roulette. Better - with the retainer, but the usual one will fit.
- Paper and pencil or pen.
- Calculator (or count in a column or in mind).
The set of tools is simple, there is in every farm. It is easier to measure with an assistant, but you can cope with yourself.
First you need to measure the length of the walls. It is advisable to do it along the walls, but if all of them are forced heavy furniture, you can carry out measurements and in the middle. Only in this case, follow the roulette tape lying along the walls, and not the defective - the measurement error will be less.
Rectangular room
If the room is the right form, without protruding parts, calculate the room area simply. Measure the length and width, write on a piece of paper. Write numbers in meters, after the comma put centimeters. For example, the length is 4.35 m (430 cm), the width is 3.25 m (325 cm).
Found numbers in short, we get the room in square meters. If we turn to our example, then the following will be: 4.35 m * 3.25 m \u003d 14,1375 square meters. m. In this magnitude, usually two digits after the semicolon are left, it means somewhat. Total, the calculated square of the room is 14.14 square meters.
Placement of irregular shape
If you need to calculate the area of \u200b\u200bthe wrong shape, it is broken into simple figures - squares, rectangles, triangles. Then they measure all the necessary dimensions, produce calculations according to the well-known formulas (there is a little lower in the table).
One example is in the photo. Since both are a rectangle, the area is considered in the same formula: the length is multiplied by the width. The number found should be taken or add to the room size - depending on the configuration.
Square Square
- We consider the square without protrusion: 3.6 m * 8.5 m \u003d 30.6 square meters. m.
- We consider the dimensions of the serving part: 3.25 m * 0.8 m \u003d 2.6 square meters. m.
- We fold two quantities: 30.6 square meters. m. + 2.6 square meters. m. \u003d 33.2 kV. m.
There are also rooms with beveled walls. In this case, we divide it so that rectangles and triangle are obtained (as in the figure below). As you can see, for a given case, you need to have five sizes. It was possible to smash differently, putting a vertical, not a horizontal line. It does not matter. A set of simple figures is simply required, and the method of their allocation is arbitrary.
In this case, the order of calculations is:
- We consider a large rectangular part: 6.4 m * 1.4 m \u003d 8.96 square meters. m. If rounded, we get 9, 0 sq.m.
- Consider a small rectangle: 2.7 m * 1.9 m \u003d 5.13 kV. m. Rounded, we get 5.1 square meters. m.
- We consider the area of \u200b\u200bthe triangle. Since it is with a straight angle, it is equal to half of the area of \u200b\u200bthe rectangle with the same dimensions. (1.3 m * 1.9 m) / 2 \u003d 1.235 square meters. m. After rounding, we get 1.2 square meters. m.
- Now we are all fold to find the total area of \u200b\u200bthe room: 9.0 + 5,1 + 1.2 \u003d 15.3 square meters. m.
The layout of the premises can be very diverse, but the general principle you understood: we divide on simple figures, we measure all the required dimensions, calculate the square of each fragment, then we add everything.
Another important note: the room, floor and ceiling area are all the same values. Differences may be if there are some semi-colones that do not reach the ceiling. Then the quadrature of these elements is deducted from the total quadrature. As a result, we get the floor area.
How to calculate the square of the walls
The determination of the walls of the walls is often required when purchasing finishing materials - wallpaper, plaster, etc. For this calculation, additional measurements are needed. The width and length of the room will be needed:
- ceiling height;
- height and width of doorways;
- height and width of window openings.
All measurements - in meters, as the square of the walls are also taken to measure in square meters.
Since the walls are rectangular, then the area is considered for a rectangle: a length is multiplied by the width. In the same way, we calculate the size of the windows and doorways, their dimensions are subtracted. For example, calculate the area of \u200b\u200bthe walls shown in the diagram above.
- Wall with door:
- 2.5 m * 5.6 m \u003d 14 kV. m. - Total area of \u200b\u200bthe long wall
- how much does the doorway take: 2.1 m * 0.9 m \u003d 1.89 sq.m.
- wall without taking into account the doorway - 14 sq.m - 1.89 square meters. m \u003d 12,11 square meters. M.
- Wall with window:
- quadrature of small walls: 2.5 m * 3.2 m \u003d 8 sq.m.
- how much it takes the window: 1.3 m * 1.42 m \u003d 1.846 kV. m, round, we get 1.75 sq.m.
- wall without window opening: 8 square meters. M - 1.75 sq. M \u003d 6.25 sq.m.
Find the total area of \u200b\u200bthe walls will not be difficult. We fold all four digits: 14 sq.m + 12.11 sq.m. + 8 sq.m + 6.25 sq.m. \u003d 40.36 square meters. m.
Room volume
For some calculations, the size of the room is required. In this case, three values \u200b\u200bare multiplied: width, length and room height. This value is measured in cubic meters (cubic meters), called more cubature. For example, use the data from the previous paragraph:
- long - 5.6 m;
- width - 3.2 m;
- height - 2.5 m.
If you all multiply, we get: 5.6 m * 3.2 m * 2.5 m \u003d 44.8 m 3. So, the size of the room is 44.8 cube.
The area of \u200b\u200bthe rectangle, how will not bother to sound, but this is an important concept. In everyday life, we are constantly faced with him. Find out the size of the fields, gardens, calculate the amount of paint required for the ceiling whitewash, how much will the wallpaper need for pasting
mnat and more.
Geometric figure
For a start, let's talk about the rectangle. This is a figure on a plane that has four straight corners, and its opposite sides are equal. The sides are used to calling long and width. Measure them in millimeters, centimeters, decimeters, meters, etc. They will now answer the question: "How to find a square of a rectangle?" To do this, multiply the length to the width.
Area \u003d Length * Width
But another reservation: the length and width must be expressed in the same measurement units, i.e., meter and meter, not meter and centimeter. The Latin Letter S. is recorded. For convenience, we denote the length of the Latin letter B, and the width of the Latin letter A, as shown in the picture. From here we conclude that the unit of measuring the area is mm 2, cm 2, m 2, etc.
Consider on a specific example, how to find a rectangle area. Length B \u003d 10 units. Width a \u003d 6 units. Solution: S \u003d A * B, S \u003d 10 units. * 6, s \u003d 60 units 2. A task. How to find out the area of \u200b\u200bthe rectangle if the length is 2 times the width and is 18 m? Solution: If B \u003d 18 m, then a \u003d b / 2, a \u003d 9 m. How to find a rectangle area if both sides are known? Right, substitute in the formula. S \u003d A * B, S \u003d 18 * 9, S \u003d 162 m 2. Answer: 162 m 2. A task. How much do you need to buy wallpaper rolls for the room, if its sizes make up: length 5.5 m width 3.5, and height 3 m? Wallpaper roll dimensions: 10 m length, width 50 cm. Solution: make a drawing of the room.
Square of opposite sides are equal. Calculate the wall area with dimensions 5.5 m and 3 m. S walls 1 \u003d 5.5 * 3,
S Walls 1 \u003d 16.5 m 2. Consequently, the opposite wall has an area equal to 16.5 m 2. Find areas of the following two walls. Thes of them, respectively, are 3.5 m and 3 m. S walls 2 \u003d 3.5 * 3, s walls 2 \u003d 10.5 m 2. It means that the opposite side is 10.5 m 2. Mix all the results. 16.5 + 16.5 + 10.5 + 10.5 \u003d 54 m 2. How to calculate the area of \u200b\u200bthe rectangle, if the parties are expressed in different units of measurement. Earlier, we calculated the square in M \u200b\u200b2, then in this case we will use meters. Then the width of the wallpaper will be 0.5 m. S roll \u003d 10 * 0.5, S roll \u003d 5 m 2. Now we learn how many rolls are needed for pasting room. 54: 5 \u003d 10.8 (rolls). Since they are measured by integers, then you need to buy 11 rolls of wallpaper. Answer: 11 rolls of wallpaper. A task. How to calculate the area of \u200b\u200bthe rectangle, if it is known that the width is 3 cm shorter than the length, and the sum of the sides of the rectangle is 14 cm? Solution: Let the length of x cm, then the width (x-3) see x + (x-3) + x + (x-3) \u003d 14, 4x-6 \u003d 14, 4x \u003d 20, x \u003d 5 cm - Length Rectangle, 5-3 \u003d 2 cm - the width of the rectangle, S \u003d 5 * 2, S \u003d 10 cm 2 Answer: 10 cm 2.
Summary
Having considered examples, I hope it became clear how to find a rectangle area. Let me remind you that the length of measurement of length and widths must match, otherwise it will be an incorrect result to prevent errors, read the task carefully. Sometimes the party can be expressed in the other side, do not fear. Contact our solved tasks, you can help. But at least once in life we \u200b\u200bare faced with the location of the rectangle area.
