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. Size garden plot will also be characterized by area. Even renovations in an apartment cannot be done without this definition. Therefore, the question of how to find the area of a rectangle is on our life path 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 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 high school, 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 the 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 right triangle. After all, if you look at it, a 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, you need to know the radius of the circle. How to calculate the area of a rectangle 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.
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 of material needed. 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 a flat figure in which opposite sides are equal and the angles are 90 degrees. To denote area in mathematics, the English letter S is used. 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 the 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, you 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.
Lesson on the topic: "Formulas for determining the area of a triangle, rectangle, square"
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Definition and concept of area of a figure
To better understand what the area of a figure is, consider the figure.This arbitrary figure is divided into 12 small squares. The side of each square is 1 cm. And the area of each square is 1 square centimeter, which is written as follows: 1 cm 2.
Then the area of the figure is 12 square centimeters. In mathematics, area is denoted Latin letter S.
This means that the area of our figure is: S shape = 12 cm 2.
The area of the figure is equal to the area of all the small squares that make it up!
Guys, remember!
Area measured square units length. Area units:
1. Square kilometer - km 2 (when the areas are very large, for example, a country or sea).
2. Square meter- m2 (quite suitable for measuring the area of a plot or apartment).
3. Square centimeter - cm 2 (usually used in mathematics lessons when drawing figures in a notebook).
4. Square millimeter - mm 2.
Area of a triangle
Let's consider two types of triangles: right-angled and arbitrary.To find the area of a right triangle, you need to know the length of the base and the height. In a right triangle, the height is replaced by one of the sides. Therefore, in the formula for the area of a triangle, instead of the height, we substitute one of the sides. 
In our example, the sides are 7 cm and 4 cm. The formula for calculating the area of a triangle is written as follows:
S of right triangle ABC = BC * CA: 2
S of right triangle ABC = 7 cm * 4 cm: 2 = 14 cm 2
Now consider an arbitrary triangle. 
For such a triangle, you need to draw the height to the base.
In our example, the height is 6 cm and the base is 8 cm. As in the previous example, we calculate the area using the formula:
S of an arbitrary triangle ABC = BC * h: 2.
Let's substitute our data into the formula and get:
S of an arbitrary triangle ABC = 8 cm * 6 cm: 2 = 24 cm 2.
Area of a rectangle and square
Take a rectangle ABCD with sides 5 cm and 8 cm.
The formula for calculating the area of a rectangle is written as follows: S rectangle ABCD = AB * BC.
S rectangle ABCD = 8 cm * 5 cm = 40 cm 2.
Now let's calculate the area of the square. Unlike a rectangle and a triangle, to find the area of a square you only need to know one side. In our example, the side of the square ABCD is 9 cm. 
S square ABCD = AB * BC = AB 2.
Let's substitute our data into the formula and get:
S square ABCD = 9 cm * 9 cm = 81 cm 2.
Lesson and presentation on the topic: "Perimeter and 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- This numerical characteristic figures. 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 summer cottage measuring 20 m by 30 m was purchased. Determine the area summer cottage, write your answer in square centimeters.
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 straight-coal-nos in order to find the area of the straight-coal-no-ka.
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 the 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
