Formula for finding the area of ​​a rectangle. Calculator for calculating the area of ​​an irregularly shaped land plot

We already got to know each other fi-gu-ry area, did you recognize one of the units from the area measurement - square centimeter. In the lesson we will teach you how to calculate the area of ​​a rectangular coal.

We already know how to find the area of ​​figures, which are times de-lined into square san-ti-meters.

For example:

We can determine that the area of ​​the first figure is 8 cm2, the area of ​​the second figure is 7 cm2.

How to find the area of ​​a rectangular corner whose sides are 3 cm and 4 cm long?

To solve the problem, let’s cut the rectangle into 4 strips of 3 cm2 each.

Then the area of ​​the rectangle will be equal to 3 * 4 = 12 cm2.

The same rectangle can be divided into 3 strips of 4 cm2 each.

Then the area of ​​the rectangle will be equal to 4*3=12 cm2.

In both cases, to find the area of ​​a rectangular angle, the numbers are not multiplied, you The exact lengths of the sides are straight-corner.

Let's find the area of ​​each straight coal.

We look at the rectangular nickname of AKMO.

There are 6 cm2 in one strip, and there are 2 such strips in this rectangle. So, we can do the following: effect:

The number 6 denotes the length of the straight-corner, and 2 means the shi-ri-well of the straight-corner. Thus, we moved through hundreds of rectangular corners in order to find the area of ​​the rectangular corner.

Consider the rectangular nickname KDCO.

In a rectangular KDCO in one strip there is 2cm2, and there are 3 such strips. Therefore, we can perform the action

The number 3 denotes the length of the straight-corner, and 2 means the shi-ri-well of the straight-corner. We re-lived a lot of them and found out the square-square area.

We can conclude: to find the area of ​​a rectangular angle, you don’t need to divide the fi-gu-ru into square san-ti-meters every time.

To calculate the area of ​​a rectangular corner, you need to find its length and shi-ri-well (the lengths of the sides of a rectangular corner must be you -same in the same units from-measurement), and then calculate the resulting numbers (flat there will be mercy in the same amount of space)

To summarize: the area of ​​a rectangular angle is equal to the product of its length and width.

Re-shi-te for-da-chu.

Can you calculate the area of ​​a rectangle, if the length of the rectangle is 9 cm, and the width is 2 cm.

Let's say we eat like this. In this case, both the length and the shi-ri-na are straight-corner. Therefore, we act according to the law: the area of ​​a rectangular angle is equal to the product of its length and width.

We are writing a decision.

Answer: rectangular area 18cm2

What other lengths do you think the sides could be of a straight angle with such an area?

You can think like this. Since the area is the product of the lengths of the sides, it is necessary to remember the table cleverly -nia. When you multiply what numbers, you get the answer 18?

That's right, when you multiply 6 and 3, you also get 18. This means that a rectangle can have sides of 6 cm and 3 cm and its area will also be equal to 18 cm2.

Re-shi-te for-da-chu.

The length of the rectangle is 8 cm, and the length is 2 cm. Find its area and perimeter.

We know the length and the shi-ri-na-straight-angle-no-ka. It is necessary to remember that in order to find an area it is necessary to find the product of its length and width , and to find the perimeter you need to multiply the sum of the length and the shi-ri by two.

We are writing a decision.

Answer: the area of ​​the rectangle is 16 cm2, and the perimeter of the rectangle is 20 cm.

Re-shi-te for-da-chu.

The length of the rectangle is 4 cm, and the length of the shi-ri-na is 3 cm. What is the area of ​​the triangle? (look ri-su-nok)

To answer the question for-da-chi, sna-cha-la, you need to find the area of ​​straight-coal-no. We know that for this it is necessary to multiply the length by shi-ri-nu.

Look at the drawing. Have you dia-go-nal divided a right-angle into two equal triangles? Next, the area of ​​one triangular corner is 2 times smaller than the area of ​​a rectangular corner. So, cheat, you need to reduce 12 by 2 times.

Answer: the area of ​​the triangle is 6 cm2.

Today, in class, we learned how to calculate the area of ​​a rectangular coal and learned how to use Take this rule when solving problems involving finding an area in a straight line.

SOURCES

http://interneturok.ru/ru/school/matematika/3-klass/tema/ploschad-pryamougolnika?seconds=0&chapter_id=1779

One of the first formulas that is studied in mathematics is related to the rectangle. It is also the most frequently used. Rectangular surfaces surround us everywhere, so we often need to know their areas. At least to find out whether the available paint is enough to paint the floors.

What units of area are there?

If we talk about the one that is accepted as international, then it will be a square meter. It is convenient to use when calculating the areas of walls, ceilings or floors. They indicate the area of ​​housing.

When it comes to smaller objects, square decimeters, centimeters or millimeters are entered. The latter are needed if the figure is no larger than a fingernail.

When measuring the area of ​​a city or country, square kilometers are the most appropriate. But there are also units that are used to indicate the size of the area: are and hectare. The first of them is also called a hundred.

What if the sides of the rectangle are given?

In a similar way, which is a special case of a rectangle is calculated. Since all sides are equal, the product becomes the square of the letter A.

What if the figure is depicted on checkered paper?

In this situation, you need to rely on the number of cells inside the figure. Using their number, it is easy to calculate the area of ​​a rectangle. But this can be done when the sides of the rectangle coincide with the lines of the cells.

Often the rectangle is positioned in such a way that its sides are inclined relative to the paper line. Then the number of cells is difficult to determine, so calculating the area of ​​the rectangle becomes more complicated.

You will first need to find out the area of ​​the rectangle, which can be drawn in cells exactly around this one. It's simple: multiply the height and width. Then subtract from the resulting area of ​​all And there are four of them. By the way, they are calculated as half the product of the legs.

The final result will give the area of ​​this rectangle.

What to do if the sides are unknown, but its diagonal and the angle between the diagonals are given?

Before that, in this situation, you need to calculate its sides in order to use the already familiar formula. First you need to remember the property of its diagonals. They are equal and bisected by the point of intersection. You can see in the drawing that the diagonals divide the rectangle into four isosceles triangles, which are equal in pairs to each other.

The equal sides of these triangles are defined as halves of the diagonal, which is known. That is, each triangle has two sides and an angle between them, which are given in the problem. You can use

One side of the rectangle will be calculated using a formula that uses the equal sides of the triangle and the cosine of the given angle. To calculate the second, the cosine value will have to be taken from the angle equal to the difference of 180 and the known angle.

What to do if the problem gives a perimeter?

Usually the condition also indicates the ratio of length and width. The question of how to calculate the area of ​​a rectangle is simpler in this case using a specific example.

Let us assume that in the problem the perimeter of a certain rectangle is 40 cm. It is also known that its length is one and a half times greater than its width. You need to find out its area.

Solving the problem begins by writing the perimeter formula. It is more convenient to write it down as the sum of length and width, each of which is multiplied by two separately. This will be the first equation in the system that needs to be solved.

The second is related to the aspect ratio known by condition. The first side, that is, the length, is equal to the product of the second (width) and the number 1.5. This equality must be substituted into the formula for the perimeter.

It turns out that it is equal to the sum of two monomials. The first is the product of 2 and an unknown width, the second is the product of the numbers 2 and 1.5 and the same width. There is only one unknown in this equation: width. You need to count it, and then use the second equality to calculate the length. All that remains is to multiply these two numbers to find out the area of ​​the rectangle.

Calculations give the following values: width - 8 cm, length - 12 cm, and area - 96 cm 2. The last number is the answer to the problem considered.

The area of ​​a rectangle may not sound arrogant, but it is an important concept. In everyday life we ​​constantly encounter it. Find out the size of fields, vegetable gardens, calculate the amount of paint needed to whitewash the ceiling, how much wallpaper will be needed for pasting

money and more.

Geometric figure

First, let's talk about the rectangle. This is a figure on a plane that has four right angles and its opposite sides are equal. Its sides are usually called length and width. They are measured in millimeters, centimeters, decimeters, meters, etc. Now we will answer the question: “How to find the area of ​​a rectangle?” To do this, you need to multiply the length by the width.

Area=length*width

But one more caveat: length and width must be expressed in the same units of measurement, that is, meter and meter, and not meter and centimeter. The area is written with the Latin letter S. For convenience, let’s denote the length with the Latin letter b, and the width with the Latin letter a, as shown in the figure. From this we conclude that the unit of area is mm 2, cm 2, m 2, etc.

Let's look at a specific example of how to find the area of ​​a rectangle. Length b=10 units. Width a=6 units. Solution: S=a*b, S=10 units*6 units, S=60 units 2. Task. How to find out the area of ​​a rectangle if the length is 2 times the width and is 18 m? Solution: if b=18 m, then a=b/2, a=9 m. How to find the area of ​​a rectangle if both sides are known? That's right, substitute it into the formula. S=a*b, S=18*9, S=162 m 2. Answer: 162 m2. Task. How many rolls of wallpaper do you need to buy for a room if its dimensions are: length 5.5 m, width 3.5, and height 3 m? Dimensions of a roll of wallpaper: length 10 m, width 50 cm. Solution: make a drawing of the room.

The areas of opposite sides are equal. Let's calculate the area of ​​a wall with dimensions of 5.5 m and 3 m. S wall 1 = 5.5 * 3,

S wall 1 = 16.5 m 2. Therefore, the opposite wall has an area of ​​16.5 m2. Let's find the area of ​​the next two walls. Their sides, respectively, are 3.5 m and 3 m. S wall 2 = 3.5 * 3, S wall 2 = 10.5 m 2. This means that the opposite side is also equal to 10.5 m2. Let's add up all the results. 16.5+16.5+10.5+10.5=54 m2. How to calculate the area of ​​a rectangle if the sides are expressed in different units of measurement. Previously, we calculated areas in m2, and in this case we will use meters. Then the width of the wallpaper roll will be equal to 0.5 m. S roll = 10 * 0.5, S roll = 5 m 2. Now we’ll find out how many rolls are needed to cover a room. 54:5=10.8 (rolls). Since they are measured in whole numbers, you need to buy 11 rolls of wallpaper. Answer: 11 rolls of wallpaper. Task. How to calculate the area of ​​a 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 be x cm, then the width is (x-3) cm. x+(x-3)+x+(x-3)=14, 4x-6=14, 4x=20, x=5 cm - length rectangle, 5-3=2 cm - width of the rectangle, S=5*2, S=10 cm 2 Answer: 10 cm 2.

Summary

Having looked at the examples, I hope it has become clear how to find the area of ​​a rectangle. Let me remind you that the units of measurement for length and width must match, otherwise you will get an incorrect result. To avoid mistakes, read the task carefully. Sometimes a side can be expressed through the other side, don't be afraid. Please refer to our solved problems, it is quite possible that they can help. But at least once in our lives we are faced with finding the area of ​​a rectangle.

L * H = S to find the area of ​​a rectangle, you need to multiply the width by the length. In other words, it can be expressed like this: The area of ​​a rectangle is equal to the product of the sides.

1. Let's give an example of calculation how to find the area of ​​a rectangle, the sides are equal to known quantities, for example width 4 cm, length 8 cm.

How to find the area of ​​a rectangle with sides 4 and 8 cm: The solution is simple! 4 x 8 = 32 cm2. To solve such a simple problem, you need to calculate the product of the sides of the rectangle or simply multiply the width by the length, this will be the area!

2. A special case of a rectangle is a square, this is the case when the sides of the rectangle are equal, in this case you can find the area of ​​the square using the above formula.

What is the area of ​​the rectangle?

The ability to calculate the area of ​​a rectangle is a basic skill for solving a huge number of everyday or technical problems. This knowledge is applied in almost all areas of life! For example, in cases where areas of any surfaces are needed in construction or real estate. When calculating areas of land, plots, walls of houses, living quarters... it is impossible to name a single area of ​​human activity where this knowledge cannot be useful!

If calculating the area of ​​a rectangle causes you difficulties - just use our calculator! O will instantly provide all the necessary calculations and write the text of the solution with explanations in detail.

Starting in grade 5, students begin to become familiar with the concept of areas of different shapes. A special role is given to the area of ​​the rectangle, since this figure is one of the easiest to study.

Area Concepts

Any figure has its own area, and the calculation of the area is based on a unit square, that is, a square with a long side of 1 mm, or 1 cm, 1 dm, and so on. The area of ​​such a figure is equal to $1*1 = 1mm^2$, or $1cm^2$, etc. The area, as a rule, is denoted by the letter – S.

The area shows the size of the part of the plane occupied by the figure outlined by the segments.

A rectangle is a quadrilateral in which all angles are of the same degree measure and equal to 90 degrees, and the opposite sides are parallel and equal in pairs.

Particular attention should be paid to the units of measurement of length and width. They must match. If the units do not match, they are converted. As a rule, they convert a larger unit into a smaller one, for example, if the length is given in dm and the width is in cm, then dm is converted to cm, and the result will be $cm^2$.

Rectangle area formula

In order to find the area of ​​a rectangle without a formula, you need to count the number of unit squares into which the figure is divided.

Rice. 1. Rectangle divided into unit squares

The rectangle is divided into 15 squares, that is, its area is 15 cm2. It is worth noting that the figure takes up 3 squares in width and 5 in length, so to calculate the number of unit squares, you need to multiply the length by the width. The smaller side of the quadrilateral is the width, the longer the length. Thus, we can derive the formula for the area of ​​a rectangle:

S = a · b, where a,b are the width and length of the figure.

For example, if the length of the rectangle is 5 cm and the width is 4 cm, then the area will be equal to 4 * 5 = 20 cm 2.

Calculating the area of ​​a rectangle using its diagonal

In order to calculate the area of ​​a rectangle through the diagonal, you need to apply the formula:

$$S = (1\over(2)) ⋅ d^2 ⋅ sin(α)$$

If the task gives the values ​​of the angle between the diagonals, as well as the value of the diagonal itself, then you can calculate the area of ​​the rectangle using the general formula for arbitrary convex quadrilaterals.

A diagonal is a line segment that connects opposite points of a figure. The diagonals of the rectangle are equal, and the point of intersection is divided in half.

Rice. 2. Rectangle with drawn diagonals

Examples

To reinforce the topic, consider examples of tasks:

No. 1. Find the area of ​​a garden plot of the same shape as in the figure.

Rice. 3. Drawing for the problem

Solution:

In order to subtract the area, you need to divide the figure into two rectangles. One of them will have dimensions of 10 m and 3 m, the other 5 m and 7 m. Separately, we find their areas:

$S_1 =3*10=30 m^2$;

This will be the area of ​​the garden plot $S = 65 m^2$.

No. 2. Subtract the area of ​​the rectangle if given its diagonal d = 6 cm and the angle between the diagonals α = 30 0.

Solution:

Value $sin 30 =(1\over(2)) $,

$ S =(1\over(2))⋅ d^2 ⋅ sinα$

$S =(1\over(2)) * 6^2 * (1\over(2)) =9 cm^2$

Thus, $S=9 cm^2$.

The diagonals divide the rectangle into 4 shapes - 4 triangles. In this case, the triangles are equal in pairs. If you draw a diagonal in a rectangle, it divides the figure into two equal right triangles. Average rating: 4.4. Total ratings received: 214.