is a parallelogram in which all angles are equal to 90°, and opposite sides are parallel and equal in pairs.
A rectangle has several irrefutable properties that are used in solving many problems, in formulas for the area of a rectangle and its perimeter. Here they are:
The length of an unknown side or diagonal of a rectangle is calculated using or using the Pythagorean theorem. The area of a rectangle can be found in two ways - by the product of its sides or by the formula for the area of a rectangle through the diagonal. The first and simplest formula looks like this:
An example of calculating the area of a rectangle using this formula is very simple. Knowing two sides, for example a = 3 cm, b = 5 cm, we can easily calculate the area of the rectangle:
We find that in such a rectangle the area will be equal to 15 square meters. cm.
Area of a rectangle through diagonals
Sometimes you need to apply the formula for the area of a rectangle through the diagonals. It requires not only finding out the length of the diagonals, but also the angle between them:
Let's look at an example of calculating the area of a rectangle using diagonals. Let a rectangle with diagonal d = 6 cm and angle = 30° be given. We substitute the data into the already known formula:
So, the example of calculating the area of a rectangle through the diagonal showed us that finding the area in this way, if an angle is given, is quite simple.
Let's look at another interesting problem that will help us stretch our brains a little.
Task: Given a square. Its area is 36 square meters. cm. Find the perimeter of a rectangle whose length of one side is 9 cm and whose area is the same as the square given above.
So we have several conditions. For clarity, let’s write them down to see all the known and unknown parameters: 
The sides of the figure are parallel and equal in pairs. Therefore, the perimeter of the figure is equal to twice the sum of the lengths of the sides:
From the formula for the area of a rectangle, which is equal to the product of the two sides of the figure, we find the length of side b
From here:
We substitute the known data and find the length of side b:
Calculate the perimeter of the figure: 
This is how, knowing a few simple formulas, you can calculate the perimeter of a rectangle, knowing its area.
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.
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 how the diagonal divided the rectangle into two equal triangles? 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 on finding the area of a rectangle.
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5. “School of Russia”: Programs for primary school. - M.: “Enlightenment”, 2011.
6. S.I.Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.
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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.
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.
Area of a polygon
We will associate the concept of area of a polygon with such a geometric figure as a square. For the unit area of a polygon we will take the area of a square with a side equal to one. Let us introduce two basic properties for the concept of area of a polygon.
Property 1: For equal polygons, their areas are equal.
Property 2: Any polygon can be divided into several polygons. In this case, the area of the original polygon is equal to the sum of the areas of all the polygons into which this polygon is divided.
Square area
Theorem 1
The area of a square is defined as the square of the length of its side.
where $a$ is the length of the side of the square.
Proof.
To prove this we need to consider three cases.
The theorem has been proven.
Area of a rectangle
Theorem 2
The area of a rectangle is determined by the product of the lengths of its adjacent sides.
Mathematically this can be written as follows
Proof.
Let us be given a rectangle $ABCD$ with $AB=b,\ AD=a$. Let's build it up to a square $APRV$, the side length of which is equal to $a+b$ (Fig. 3).
Figure 3.
By the second property of areas we have
\ \ \
By Theorem 1
\ \
The theorem has been proven.
Sample tasks
Example 1
Find the area of a rectangle with sides $5$ and $3$.
A rectangle is a special case of a quadrilateral. This means that the rectangle has four sides. Its opposite sides are equal: for example, if one of its sides is 10 cm, then the opposite side will also be equal to 10 cm. A special case of a rectangle is a square. A square is a rectangle with all sides equal. To calculate the area of a square, you can use the same algorithm as to calculate the area of a rectangle.
How to find out the area of a rectangle based on two sides
In order to find the area of a rectangle, you need to multiply its length by its width: Area = Length × Width. In the case given below: Area = AB × BC.
How to find out the area of a rectangle by side and diagonal length
Some problems require you to find the area of a rectangle using the length of the diagonal and one of the sides. The diagonal of a rectangle divides it into two equal right triangles. Therefore, we can determine the second side of the rectangle using the Pythagorean theorem. After this, the task is reduced to the previous point.
How to find out the area of a rectangle by its perimeter and side
The perimeter of a rectangle is the sum of all its sides. If you know the perimeter of the rectangle and one side (such as the width), you can calculate the area of the rectangle using the following formula:
Area = (Perimeter×width – width^2)/2.
Area of a rectangle through the sine of the acute angle between the diagonals and the length of the diagonal
The diagonals in a rectangle are equal, so to calculate the area based on the length of the diagonal and the sine of the acute angle between them, you should use the following formula: Area = Diagonal^2 × sin(acute angle between the diagonals)/2.
