With such a concept as the area, we have to deal with in our lives every day. 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 the area. Even repairs in an apartment cannot be done without this definition. Therefore, the question of how to find the area of \u200b\u200ba rectangle arises very often on ours and is important not only for schoolchildren.
For those who don't know, a rectangle is a flat figure with opposite sides equal and angles 90 degrees. To denote the area in mathematics, the English letter S is used. It is measured in square units: meters, centimeters, and so on.
Now let's 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 are faced with a way to determine the area using width and length.
Let's take a rectangle with width b and length k. To calculate the area of a given rectangle, multiply the width by the length. All this can be represented as a formula that will look like this: S = b * k.
Now let's look at this method with a specific example. It is necessary to determine the area of \u200b\u200bthe 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 the width of the rectangle. At the same time, 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 need to remember the area. After all, if you figure it out, the rectangle consists of two equal right triangles. So, back to the defined 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 already using the definition of the sine, you can determine the length of the second side. How to find the area of a rectangle now? Yes, it is very simple to multiply the obtained values.
In formula form, it would 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 a circle inscribed in it. It applies 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 number 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 trick, we multiply both sides and get the area of the rectangle. In formula form, it would 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 found in life, but they can be useful for solving many problems at school. Perhaps this article will be useful for solving your problems.
Periodically, we need to know the area and volume of the room. This data may be needed when designing heating and ventilation, when purchasing building materials, and in many other situations. It is also periodically required to know the area of \u200b\u200bthe walls. All these data are calculated easily, but first you have to work with a tape measure - measure all the required dimensions. How to calculate the area of \u200b\u200bthe room and walls, the volume of the room and will be discussed further.
Room area in square meters
- Roulette. Better - with a latch, but a regular one will do.
- Paper and pencil or pen.
- Calculator (or count in a column or in your head).
A set of tools is simple, there is in every household. It is easier to measure with an assistant, but you can do it yourself.
First you need to measure the length of the walls. It is desirable to do this along the walls, but if they are all full of heavy furniture, you can take measurements in the middle. Only in this case, make sure that the tape measure lies along the walls, and not obliquely - the measurement error will be less.
Rectangular room
If the room is of the correct shape, without protruding parts, it is easy to calculate the area of \u200b\u200bthe room. Measure the length and width, write it down on a piece of paper. Write the numbers in meters, put centimeters after the decimal point. For example, length 4.35 m (430 cm), width 3.25 m (325 cm).
We multiply the found numbers, we get the area of \u200b\u200bthe room in square meters. If we turn to our example, we get the following: 4.35 m * 3.25 m = 14.1375 sq. m. In this value, usually two digits after the decimal point are left, which means we round off. In total, the calculated quadrature of the room is 14.14 square meters.
Irregular room
If you need to calculate the area of a room of irregular shape, it is divided into simple shapes - squares, rectangles, triangles. Then they measure all the necessary dimensions, make calculations according to known formulas (there is in the table just below).
One example is in the photo. Since both are rectangles, the area is calculated using the same formula: multiply the length by the width. The figure found must be subtracted or added to the size of the room, depending on the configuration.
Complex room area
- We consider the quadrature without a ledge: 3.6 m * 8.5 m = 30.6 sq. m.
- We consider the dimensions of the protruding part: 3.25 m * 0.8 m = 2.6 sq. m.
- We add two values: 30.6 square meters. m. + 2.6 sq. m. = 33.2 sq. m.
There are also rooms with sloping walls. In this case, we split it so that we get rectangles and a triangle (as in the figure below). As you can see, for this case it is required to have five sizes. It could have been split differently by putting a vertical rather than a horizontal line. It does not matter. It just requires a set of simple shapes, and the way they are selected is arbitrary.
In this case, the calculation order is:
- We consider the large rectangular part: 6.4 m * 1.4 m \u003d 8.96 square meters. m. If we round up, we get 9.0 sq.m.
- We calculate a small rectangle: 2.7 m * 1.9 m \u003d 5.13 square meters. m. Rounding up, we get 5.1 square meters. m.
- We calculate the area of the triangle. Since it is with a right angle, it is equal to half the area of a rectangle with the same dimensions. (1.3 m * 1.9 m) / 2 = 1.235 sq. m. After rounding, we get 1.2 square meters. m.
- Now we add everything up to find the total area of the room: 9.0 + 5.1 + 1.2 \u003d 15.3 square meters. m.
The layout of the premises can be very diverse, but you understand the general principle: we divide into simple figures, measure all the required dimensions, calculate the quadrature of each fragment, then add everything up.
Another important note: the area of \u200b\u200bthe room, floor and ceiling are all the same values. Differences can be if there are some semi-columns that do not reach the ceiling. Then the quadrature of these elements is subtracted from the total quadrature. The result is the floor area.
How to calculate the square of the walls
Determining the area of walls is often required when purchasing finishing materials - wallpaper, plaster, etc. This calculation requires additional measurements. To the already existing width and length of the room you will need:
- ceiling height;
- height and width of doorways;
- height and width of window openings.
All measurements are in meters, since the square of the walls is also usually measured in square meters.
Since the walls are rectangular, the area is calculated as for a rectangle: we multiply the length by the width. In the same way, we calculate the dimensions of windows and doorways, subtract their dimensions. For example, we calculate the area of \u200b\u200bthe walls shown in the diagram above.
- Wall with a door:
- 2.5 m * 5.6 m = 14 square meters m. - the total area of \u200b\u200bthe long wall
- how much does a doorway take: 2.1 m * 0.9 m = 1.89 sq.m.
- wall excluding doorway - 14 sq.m - 1.89 sq.m. m = 12.11 sq. m
- Wall with a window:
- square of small walls: 2.5 m * 3.2 m = 8 sq.m.
- how much does a window take up: 1.3 m * 1.42 m = 1.846 sq. m, rounding up, we get 1.75 sq.m.
- wall without a window opening: 8 sq. m - 1.75 sq.m = 6.25 sq.m.
Finding the total area of the walls is not difficult. We add up all four numbers: 14 sq.m + 12.11 sq.m. + 8 sq.m. + 6.25 sq.m. = 40.36 sq. m.
Room volume
Some calculations require the volume of the room. In this case, three values are multiplied: width, length and height of the room. This value is measured in cubic meters (cubic meters), also called cubic capacity. For example, we use the data from the previous paragraph:
- length - 5.6 m;
- width - 3.2 m;
- height - 2.5 m.
If we multiply everything, we get: 5.6 m * 3.2 m * 2.5 m = 44.8 m 3. So, the volume of the room is 44.8 cubic meters.
Lesson on the topic: "Formulas for determining the area of a triangle, rectangle, square"
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Teaching aids and simulators in the online store "Integral" for grade 5
Simulator for the textbook by I.I. Zubareva and A.G. Mordkovich
Simulator for the textbook by G.V. Dorofeev and L.G. Peterson
Definition and concept of the area of \u200b\u200ba figure
To better understand what the area of \u200b\u200bthe 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 cm2.
Then the area of the figure is 12 square centimeters. In mathematics, area is denoted by the Latin letter S.
So the area of our figure is: S figures \u003d 12 cm 2.
The area of the figure is equal to the area of all the small squares of which it consists!
Guys, remember!
Area is measured in square units of length. Area units:
1. Square kilometer - km 2 (when the areas are very large, for example, a country or a sea).
2. Square meter - m 2 (quite suitable for measuring the area of \u200b\u200ba 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
Consider two types of triangles: rectangular 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, one of the sides replaces the height. 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 * SA: 2
S of a right triangle ABC \u003d 7 cm * 4 cm: 2 \u003d 14 cm 2
Now consider an arbitrary triangle. 
For such a triangle, it is necessary 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.
Substitute our data into the formula and get:
S of an arbitrary triangle ABC \u003d 8 cm * 6 cm: 2 \u003d 24 cm 2.
Area of rectangle and square
Take a rectangle ABCD with sides 5 cm and 8 cm.
The formula for calculating the area of a rectangle is: S rectangle ABCD = AB * BC.
S rectangle ABCD \u003d 8 cm * 5 cm \u003d 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 need to know only one side. In our example, the side of the square ABCD is 9 cm. 
S of the square ABCD \u003d AB * BC \u003d AB 2.
Substitute our data into the formula and get:
S square ABCD \u003d 9 cm * 9 cm \u003d 81 cm 2.
With such a concept as the area, we have to deal with in our lives every day. 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 the area. Even repairs in an apartment cannot be done without this definition. Therefore, the question of how to find the area of a rectangle arises very often on our life path and is important not only for schoolchildren.
For those who don't know, a rectangle is a flat figure with opposite sides equal and angles 90°. To denote the area in mathematics, the English letter S is used. It is measured in square units: meters, centimeters, and so on.
Now let's 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 are faced with a way to determine the area using width and length.
Let's take a rectangle with width b and length k. To calculate the area of a given rectangle, multiply the width by the length. All this can be represented in the form of a formula that will look like this: S \u003d b * k
Now let's look at this method with a specific example. It is necessary to determine the area of \u200b\u200bthe 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 the width of the rectangle. At the same time, 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 need to remember the area of the right triangle. After all, if you look, then the rectangle consists of two equal right-angled triangles. So, back to the defined 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 already using the definition of the sine, you can determine the length of the second side. How to find the area of a rectangle now? Yes, it is very simple to multiply the obtained values.
In formula form, it would 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 a circle inscribed in it. It applies 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 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 number 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 trick, we multiply both sides and get the area of the rectangle. In formula form, it would 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 found in life, but they can be useful for solving many problems at school. Perhaps this article will be useful for solving your problems.
A useful calculator for schoolchildren and adults allows you to quickly calculate the area of a rectangle on its two sides. We often make such a calculation not only as part of a school geometry course, but also in everyday life. For example, if you need to calculate the area of a room when repairing an apartment, to calculate the required amount of materials.
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