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 we're talking about about smaller objects, then enter square decimeters, centimeters or millimeters. 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 triangle, which are pairwise equal 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 the formula that includes equal sides triangle and the cosine of a 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.

We have to deal with such a concept as area in our daily lives. So, for example, when building a house you need to know it in order to calculate the amount required material. The size of the garden plot will also be characterized by its area. Even renovations in an apartment cannot be done without this definition. Therefore, the question of how to find the area of ​​a rectangle comes up very often and is important not only for schoolchildren.

For those who don't know, a rectangle is flat figure, in which opposite sides are equal and the angles are 90°. To denote area in mathematics we use English letter S. It is measured in square units: meters, centimeters, and so on.

Now we will try to give a detailed answer to the question of how to find the area of ​​a rectangle. There are several ways to determine this value. Most often we come across a method of determining area using width and length.

Let's take a rectangle with width b and length k. To calculate the area of ​​a given rectangle, you need to multiply the width by the length. All this can be represented in the form of a formula that will look like this: S = b * k.

Now let's look at this method using a specific example. It is necessary to determine the area of ​​a garden plot with a width of 2 meters and a length of 7 meters.

S = 2 * 7 = 14 m2

In mathematics, especially in mathematics, we have to determine the area in other ways, since in many cases we do not know either the length or width of the rectangle. At the same time, other known quantities exist. How to find the area of ​​a rectangle in this case?

• If we know the length of the diagonal and one of the angles that makes up the diagonal with any side of the rectangle, then in this case we will need to remember the area. After all, if you look at it, the rectangle consists of two equal right triangles. So, let's return to the determined value. First you need to determine the cosine of the angle. Multiply the resulting value by the length of the diagonal. As a result, we get the length of one of the sides of the rectangle. Similarly, but using the definition of sine, you can determine the length of the second side. How to find the area of ​​a rectangle now? Yes, it’s very simple, multiply the resulting values.

In formula form it will look like this:

S = cos(a) * sin(a) * d2, where d is the length of the diagonal

• Another way to determine the area of ​​a rectangle is through the circle inscribed in it. It is used if the rectangle is a square. To use this method need to know How to calculate the area of ​​a rectangle in this way? Of course, according to the formula. We will not prove it. And it looks like this: S = 4 * r2, where r is the radius.

It happens that instead of the radius, we know the diameter of the inscribed circle. Then the formula will look like this:

S=d2, where d is the diameter.

• If one of the sides and the perimeter are known, then how to find out the area of ​​the rectangle in this case? To do this, you need to make a series of simple calculations. As we know, the opposite sides of a rectangle are equal, so the known length multiplied by two must be subtracted from the perimeter value. Divide the result by two and get the length of the second side. Well, then the standard technique is to multiply both sides and get the area of ​​the rectangle. In formula form it will look like this:

S=b* (P - 2*b), where b is the length of the side, P is the perimeter.

As you can see, the area of ​​a rectangle can be determined in various ways. It all depends on what quantities we know before considering this issue. Of course, the latest calculus methods are practically never encountered in life, but they can be useful for solving many problems in school. Perhaps this article will be useful for solving your problems.

The area of ​​a rectangle may not sound arrogant, but it is an important concept. IN everyday life we are constantly faced with 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, not meter and centimeter. The area is recorded Latin letter S. For convenience, let’s denote the length by the Latin letter b, and the width by 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 measurements. Previously, we calculated areas in m2, then 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.

Resume

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, they may be able to help. But at least once in our lives we are faced with finding the area of ​​a rectangle.

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What are rectangle and square

Rectangle is a quadrilateral with all right angles. This means that opposite sides are equal to each other.

Square is a rectangle with equal sides and equal angles. It is called a regular quadrilateral.

Quadrangles, including rectangles and squares, are designated by 4 letters - vertices. Latin letters are used to designate vertices: A, B, C, D...

Example.

It reads like this: quadrilateral ABCD; square EFGH.

What is the perimeter of a rectangle? Formula for calculating perimeter

Perimeter of a rectangle is the sum of the lengths of all sides of the rectangle or the sum of the length and width multiplied by 2.

The perimeter is indicated by a Latin letter P. Since the perimeter is the length of all sides of the rectangle, the perimeter is written in units of length: mm, cm, m, dm, km.

For example, the perimeter of rectangle ABCD is denoted as P ABCD, where A, B, C, D are the vertices of the rectangle.

Let's write down the formula for the perimeter of a quadrilateral ABCD:

P ABCD = AB + BC + CD + AD = 2 * AB + 2 * BC = 2 * (AB + BC)

Example.
Given a rectangle ABCD with sides: AB=CD=5 cm and AD=BC=3 cm.
Let's define P ABCD.

Solution:
1. Let's draw a rectangle ABCD with the original data.
2. Let’s write a formula to calculate the perimeter of a given rectangle:

P ABCD = 2 * (AB + BC)

P ABCD = 2 * (5 cm + 3 cm) = 2 * 8 cm = 16 cm

Answer: P ABCD = 16 cm.

Formula for calculating the perimeter of a square

We have a formula for determining the perimeter of a rectangle.

P ABCD = 2 * (AB + BC)

Let's use it to determine the perimeter of a square. Considering that all sides of the square are equal, we get:

P ABCD = 4 * AB

Example.
Given a square ABCD with a side equal to 6 cm. Let us determine the perimeter of the square.

Solution.
1. Let's draw a square ABCD with the original data.

2. Let us recall the formula for calculating the perimeter of a square:

P ABCD = 4 * AB

3. Let’s substitute our data into the formula:

P ABCD = 4 * 6 cm = 24 cm

Answer: P ABCD = 24 cm.

Problems to find the perimeter of a rectangle

1. Measure the width and length of the rectangles. Determine their perimeter.

2. Draw a rectangle ABCD with sides 4 cm and 6 cm. Determine the perimeter of the rectangle.

3. Draw a square SEOM with a side of 5 cm. Determine the perimeter of the square.

Where is the calculation of the perimeter of a rectangle used?

1. A plot of land has been given; it needs to be surrounded by a fence. How long will the fence be?

In this task, it is necessary to accurately calculate the perimeter of the site so as not to buy excess material for building a fence.

2. Parents decided to renovate the children's room. You need to know the perimeter of the room and its area in order to correctly calculate the amount of wallpaper.
Determine the length and width of the room in which you live. Determine the perimeter of your room.

What is the area of ​​a rectangle?

Square is a numerical characteristic of a figure. Area is measured in square units of length: cm 2, m 2, dm 2, etc. (centimeter squared, meter squared, decimeter squared, etc.)
In calculations it is denoted by a Latin letter S.

To determine the area of ​​a rectangle, multiply the length of the rectangle by its width.
The area of ​​the rectangle is calculated by multiplying the length of the AC by the width of the CM. Let's write this down as a formula.

S AKMO = AK * KM

Example.
What is the area of ​​rectangle AKMO if its sides are 7 cm and 2 cm?

S AKMO = AK * KM = 7 cm * 2 cm = 14 cm 2.

Answer: 14 cm 2.

Formula for calculating the area of ​​a square

The area of ​​a square can be determined by multiplying the side by itself.

Example.
In this example, the area of ​​a square is calculated by multiplying the side AB by the width BC, but since they are equal, the result is multiplying the side AB by AB.

S ABCO = AB * BC = AB * AB

Example.
Determine the area of ​​a square AKMO with a side of 8 cm.

S AKMO = AK * KM = 8 cm * 8 cm = 64 cm 2

Answer: 64 cm 2.

Problems to find the area of ​​a rectangle and square

1. Given a rectangle with sides 20 mm and 60 mm. Calculate its area. Write your answer in square centimeters.

2. A dacha plot measuring 20 m by 30 m was purchased. Determine the area of ​​the dacha plot and write the answer in square centimeters.

We have already become familiar with the concept area of ​​the figure, learned one of the units of area measurement - square centimeter . In this lesson we will derive a rule on how to calculate the area of ​​a rectangle.

We already know how to find the area of ​​figures that are divided into square centimeters.

For example:

We can determine that the area of ​​the first figure is 8 cm 2, the area of ​​the second figure is 7 cm 2.

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

To solve the problem, we divide the rectangle into 4 strips of 3 cm 2 each.

Then the area of ​​the rectangle will be equal to 3 * 4 = 12 cm 2.

The same rectangle can be divided into 3 strips of 4 cm 2 each.

Then the area of ​​the rectangle will be equal to 4 * 3 = 12 cm 2.

In both cases To find the area of ​​a rectangle, the numbers expressing the lengths of the sides of the rectangle are multiplied.

Let's find the area of ​​each rectangle.

Consider the rectangle AKMO.

There are 6 cm 2 in one strip, and there are 2 such strips in this rectangle. This means that we can perform the following action:

The number 6 represents the length of the rectangle, and 2 represents the width of the rectangle. So we multiplied the sides of the rectangle to find the area of ​​the rectangle.

Consider the rectangle KDCO.

In the rectangle KDCO there are 2 cm 2 in one strip, and there are 3 such strips. Therefore, we can perform the action

The number 3 denotes the length of the rectangle, and 2 the width of the rectangle. We multiplied them and found out the area of ​​the rectangle.

We can conclude: To find the area of ​​a rectangle, you do not need to divide the figure into square centimeters each time.

To calculate the area of ​​a rectangle, you need to find its 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 resulting numbers (the area will be expressed in the corresponding units of area)

Let's summarize: The area of ​​a rectangle is equal to the product of its length and width.

Solve the problem.

Calculate the area of ​​a rectangle if the length of the rectangle is 9 cm and the width is 2 cm.

Let's think like this. In this problem, both the length and width of the rectangle are known. Therefore, we follow the rule: the area of ​​a rectangle is equal to the product of its length and width.

Let's write down the solution.

Answer: rectangle area 18cm 2

What other lengths of the sides of a rectangle with such an area do you think?

You can think like this. Since area is the product of the lengths of the sides of a rectangle, you need to remember the multiplication table. What numbers are multiplied to give 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 cm 2.

Solve the problem.

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

We know the length and width of the rectangle. It is necessary to remember that to find the area you need to find the product of its length and width, and to find the perimeter you need to multiply the sum of the length and width by two.

Let's write down the solution.

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

Solve the problem.

The length of the rectangle is 4 cm, and the width is 3 cm. What is the area of ​​the triangle? (see picture)

To answer the question in the problem, you first need to find the area of ​​the rectangle. We know that for this we need to multiply the length by the width.

Look at the drawing. Did you notice that the diagonal divides the rectangle into two? equal triangle? Therefore, the area of ​​one triangle is 2 times less than the area of ​​a rectangle. This means that 12 needs to be halved.

Answer: The area of ​​the triangle is 6 cm 2.

Today in class we learned about the rule for calculating the area of ​​a rectangle and learned to apply this rule when solving problems to find the area of ​​a rectangle.

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6. S.I.Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.

7. V.N.Rudnitskaya. Tests. M., “Exam”, 2012 (127 pp.)

2. Publishing house "Prosveshcheniye" ()

1. The length of the rectangle is 7 cm, width is 4 cm. Find the area of ​​the rectangle.

2. The side of the square is 5 cm. Find the area of ​​the square.

3. Draw possible options for rectangles with an area of ​​18 cm 2.

4. Create an assignment on the topic of the lesson for your friends.