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 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 a flat figure that has 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 area 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. However, there are other known quantities. 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 of the 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. For 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, it is necessary to make a series 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 different 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 rectangular triangle ABC= BC * SA: 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 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.
A useful calculator for schoolchildren and adults that allows you to quickly calculate the area of a rectangle based on its two sides. We often make such calculations not only within school course geometry, but also in Everyday life. For example, if you need to calculate the area of a room when renovating an apartment, to calculate required quantity materials.
Convenient navigation through the article:
Rectangle area calculator
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 that the diagonal divides the rectangle into two? equal triangle? Therefore, the area of one triangle is 2 times less area 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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2. M.I.Moro, M.A.Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. M., “Enlightenment”, 2012.
3. M.I.Moro. Math lessons: Guidelines for the teacher. 3rd grade. - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. M., “Enlightenment”, 2011.
5. “School of Russia”: Programs for primary school. - M.: “Enlightenment”, 2011.
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 rectangles with an area of 18 cm 2.
4. Create an assignment on the topic of the lesson for your friends.
Starting any project, you need to know how to calculate the area of a room. This knowledge will help you choose correct amount materials, as well as correctly calculate the dimensions. Such knowledge is also necessary when you need to check the area indicated in the documents.
Our review will help you remember school knowledge and will tell you how to apply them in practice. In addition, you can use a special calculator.
Important information! When performing calculations for a room with projections, columns and niches, it is necessary to take into account their location and configuration.
How to calculate the area of a triangle
Let's figure out how to calculate the area of a triangle. If indoors irregular shape there is an element in the form right angle, then you can apply the following formula:
S=(a*b)/2;
There are other calculation options. To calculate the area of a triangle, you can first determine the quadrature of the rectangle, and then calculate the area of the triangular angle. Then the value of the triangular part is subtracted from the squaring of the rectangle.
If there is no right angle, then you can use Heron’s formula:
S=√p(p-a)(p-b)(p-c);
Wherein a, b, c represent the sides of the triangle, and R– semi-perimeter. To find out, all sides should be summed up and then divided by 2.
To calculate the area of a given figure, you can multiply the length of the base by the height drawn from this base, and then divide the result by 2.
How to find the area of a rectangle
Often there are rooms that look like a rectangle. To calculate the area of a rectangle online calculator is considered the best option. But simple calculations You can also make it yourself. There should be no protruding parts. Measure the width and length. Entries must be made in meters.
To calculate the area of a rectangle, you need to multiply the obtained values. After the decimal point, you can leave only two digits and round the rest.
The question often arises of how to calculate the area of a rectangle if there are niches. In this case, the space is divided into several figures, and they are counted separately. There is another formula for calculating area. In this case, the perimeter of the room must be multiplied by the height. To calculate the perimeter, you need to add the length and width, and then multiply them by two.
How to calculate the area of a trapezoid
To calculate the area of a trapezoid, it is important to know the lengths of two parallel sides, as well as height. In this case, the height is the perpendicular between the sides.
Here's a formula you can use:
S=1/2(A+B)*h;
It is worth considering that nothing will change depending on the angles of the sides. But it is recommended to measure the height in several places. This approach will ensure that the two sides are parallel.
Knowledge of this formula will also be useful when calculating the areas of rooms with polygonal configurations. For example, an unusual space can be divided into a rectangle and a trapezoid.
There is also a special calculator for calculating rectangular and trapezoidal rooms. In this case, the difference is that the rectangle will have the same lengths. If necessary, adjustments can be made to existing protrusions and niches.
To determine the perimeter, you need to measure the lengths of the four sides and then add them up.
How to calculate the area of a circle
Sometimes it is necessary to calculate the area of a circle, since there may be separate areas of rounded configurations in the room. Such knowledge will also be useful when calculating arched door or window openings. Let's look at shapes based on a circle. The most common are segments, quarter circles or semicircles.
Such calculations are necessary if there is a bay window or a combination of a balcony and a room. In this case, the value of the protruding circle is first determined, and then the resulting indicator is added to the rest of the area.
To calculate the area of a circle, use the following formula:
S=πR²/2,
π equals 3.14. A R² is the radius of the circle squared.
Correct calculation of sector and segment area
A special formula for the area of a sector of a circle is included in the calculator for calculations. This formula looks like this:
S= R²*arcsin(c/2R)-0.25*C*√(4 R²-C²);
How to calculate the area of a room: calculation method by floor
You can calculate the floor area. This procedure consists of the following stages. First of all, it is necessary to clear the walls of the room. It is better to take measurements in an empty room. If the room is rectangular, then you can multiply the two sides. In reality, the sides may differ slightly, which is why it is important to measure all sides. In some cases, the room may not be configured correctly. In this situation, the entire space is divided into separate rectangles. In this case, you can draw a diagram with all dimensions. Then the area of individual plots is calculated. By the way, the room will not necessarily consist only of rectangles. It can contain triangles and even circles.
If you don't know how to count square meters rooms, you can use a special calculator. When calculating dimensions, it is not necessary to observe all dimensions down to the centimeter. Rounding of values is often used. Sometimes it is necessary to take into account various recesses and protrusions in the walls.
When making calculations, it is important to take into account their purpose. If you need to know the values for installation, then you can ignore the space occupied by heavy furniture.
There are options when used indoors different levels floor. IN similar situation It is also required to divide the room into separate zones. You should not take measurements on the wall, as it may have a curved surface.
It is difficult to determine the area based on the floor, which has various protrusions in the form of waves or semicircles.
For your information! If the room is L-shaped, then in addition to two rectangular shapes it also contains a triangle. To determine its area, you can multiply one leg by the other.
How to calculate wall area
To determine the area of the walls, you need to know how to calculate the perimeter of the room. Determining the area is often required to purchase all kinds of finishing materials. In addition to the length and width of the room, you will need to find out the height of the room, as well as the dimensions of the door and window openings.
When making calculations, it is worth considering that the height of the ceilings even in the same room can vary, especially if. The values of all walls are calculated separately, and then window and door openings are subtracted from them.
Exact values can be identified using a calculator for calculating the area of the walls of a room. Sometimes it is necessary to determine the volume of a room. In such a situation, it is necessary to multiply three values: the length, width and height of the room. This indicator measured in cubic meters.
If the room has multi-level zoning, then the following calculations can be made:
- measure all sections of the walls. It is recommended to take measurements closer to the ceiling;
- calculate area indicators by floor;
- on each vertical section it is necessary to measure the length and height, and then multiply these values;
- add the indicators to the gender value.
Each room has optimal parameters that can be taken into account when planning a construction site. The minimum residential area is 8 square meters. m. Room general purpose, such as a hall or living room can have an area of 14 to 25 square meters. m. The minimum room height is 240 cm. If the ceiling is more than three meters, then such a space is more difficult to heat. The kitchen must have an area of at least 6 square meters. m.
Correct calculation room area is a guarantee quality implementation. This approach will minimize the cost of purchasing materials.
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