home Audience 1 The area of the rectangle is equal. Area of a rectangle

The area of the rectangle is equal. Area of a rectangle

Area of a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited by the closed contour of this figure). The size of the area is expressed by the number of square units contained in it.

Triangle area formulas

Formula for the area of a triangle by side and height
Area of a triangle equal to half the product of the length of a side of a triangle and the length of the altitude drawn to this side
Formula for the area of a triangle based on three sides and the radius of the circumcircle
Formula for the area of a triangle based on three sides and the radius of the inscribed circle
Area of a triangle is equal to the product of the semi-perimeter of the triangle and the radius of the inscribed circle.

Square area formulas

Formula for the area of a square by side length
Square area equal to the square of the length of its side.
Formula for the area of a square along the diagonal length
Square area equal to half the square of the length of its diagonal.
S= 1 2
2

Rectangle area formula

Area of a rectangle

Parallelogram area formulas

Formula for the area of a parallelogram based on side length and height
Area of a parallelogram
Formula for the area of a parallelogram based on two sides and the angle between them
Area of a parallelogram is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.
a b sin α

Formulas for the area of a rhombus

Formula for the area of a rhombus based on side length and height
Area of a rhombus is equal to the product of the length of its side and the length of the height lowered to this side.
Formula for the area of a rhombus based on side length and angle
Area of a rhombus is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus.
Formula for the area of a rhombus based on the lengths of its diagonals
Area of a rhombus equal to half the product of the lengths of its diagonals.

Trapezoid area formulas

Heron's formula for trapezoid
Where S is the area of the trapezoid,
- lengths of the bases of the trapezoid,
- lengths of the sides of the trapezoid,

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 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 renovating an apartment, to calculate the required amount of materials.

Convenient navigation through the article:

Rectangle area calculator

How to calculate the area of a rectangle

A rectangle is usually called a geometric flat figure that has parallel opposite sides at angles of 90 degrees. The length of this geometric figure is the value that has the largest side. In this case, the width is taken to be the size of the smaller side. To correctly calculate the area of a rectangle, you need to know certain parameters that this figure has. Among them:

diagonal;
width;
the so-called angle of inclination to one of the sides of the diagonal;
length.

Thus, you can calculate the area of a rectangle in various ways. It all depends on the amount of information about the figure, namely, what quantities we know exactly.

How to calculate the area of a rectangle given the linear parameters of its sides?

Let's use the letter “a” to denote the length of the rectangle, the letter “b” to denote its width, and the area of the geometric figure will be denoted by the letter “S”. According to this, our formula will look like this: S = a x b.

Knowing the linear parameters of a rectangle, you can easily determine its perimeter using the following formula: P = 2(a + b), where we use the letter “P” to denote the perimeter.

How can you calculate the area of a rectangle, knowing the size of one of its sides and the diagonal?

As we know, the diagonal divides any rectangle into two so-called right triangles. Let's assign the index "c" to the diagonal, and denote the length of the side with the letter "a". Now you need to perform the following procedure:

First you need to find the length of the unknown side. To do this, we use the Pythagorean formula: b = V c2 – a2.
After this, we should determine the area of our main geometric figure (rectangle) as the area of the triangle multiplied by two: Str = ½ (a x b).
According to the calculation scheme described above, the area of the rectangle in this case will be equal to: S = 2 x Str = a x b.

If we know the perimeter of the rectangle, as well as the length of one of its sides, then the area of this geometric figure can be calculated by determining the length of the second side (b = (P – 2xa), using the following formula: S = a x b.

If we know the size of the diagonal of the rectangle, as well as the angle between the side and the diagonal itself, then the area can be calculated using the following trigonometric functions: Str = ½ x c2 x sinФ x cosФ.

In this case, the total area in this case will be S = 2 x Str.

Area of a rectangle. Video lesson.

From time to time we need to know the area and volume of a 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 the walls. All this data can be easily calculated, but first you will have to work with a tape measure to measure all the required dimensions. How to calculate the area of the room and walls, the volume of the room will be discussed further.

Room area in square meters

Roulette. It’s better with a lock, but a regular one will do.
Paper and pencil or pen.
Calculator (or count in a column or in your head).

A simple set of tools can be found in every household. It’s easier to take measurements with an assistant, but you can do it yourself.

First you need to measure the length of the walls. It is advisable to do this along the walls, but if they are all filled with 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 diagonally - 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 the room. Measure the length and width and write it down on a piece of paper. Write the numbers in meters, followed by centimeters after the decimal point. For example, length 4.35 m (430 cm), width 3.25 m (325 cm).

We multiply the found numbers to get the area of the room in square meters. If we look at our example, we get the following: 4.35 m * 3.25 m = 14.1375 sq. m. In this value, usually two digits are left after the decimal point, which means we round. In total, the calculated square footage of the room is 14.14 square meters.

Irregularly shaped room

If you need to calculate the area of an irregularly shaped room, it is divided into simple shapes - squares, rectangles, triangles. Then they measure all the required dimensions and make calculations using known formulas (found 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 found figure must be subtracted or added to the size of the room - depending on the configuration.

Room area of complex shape

We calculate the quadrature without the protrusion: 3.6 m * 8.5 m = 30.6 sq. m.
We calculate the dimensions of the protruding part: 3.25 m * 0.8 m = 2.6 sq. m.
We add two values: 30.6 sq. m. + 2.6 sq. m. = 33.2 sq. m.

There are also rooms with sloping walls. In this case, we divide it so that we get rectangles and a triangle (as in the figure below). As you can see, for this case you need to have five sizes. It could have been broken differently by placing a vertical rather than a horizontal line. It doesn't matter. It just requires a set of simple shapes, and the way to select them is arbitrary.

In this case, the order of calculations is as follows:

We consider the large rectangular part: 6.4 m * 1.4 m = 8.96 sq. m. If we round, we get 9.0 sq.m.
We calculate a small rectangle: 2.7 m * 1.9 m = 5.13 sq. m. Round up, we get 5.1 sq. m.
Calculate the area of the triangle. Since it is at 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 sq. m.
Now we add everything up to find the total area of the room: 9.0 + 5.1 + 1.2 = 15.3 square meters. m.

The layout of the premises can be very diverse, but you understand the general principle: divide it into simple shapes, measure all the required dimensions, calculate the square footage of each fragment, then add everything up.

Another important note: the area of the room, floor and ceiling are all the same measurements. There may be differences 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 footage of walls

Determining the area of walls is often required when purchasing finishing materials - wallpaper, plaster, etc. This calculation requires additional measurements. In addition to the 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 footage of 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 sizes of windows and doorways, subtract their dimensions. For example, let's calculate the area of the walls shown in the diagram above.

Wall with door:
- 2.5 m * 5.6 m = 14 sq. m. - total area of the long wall
- how much does a doorway take up: 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 window:
1. squaring of small walls: 2.5 m * 3.2 m = 8 sq.m.
2. how much does a window take: 1.3 m * 1.42 m = 1.846 sq. m, round up, we get 1.75 sq.m.
3. wall without window opening: 8 sq. m - 1.75 sq.m = 6.25 sq.m.

Finding the total area of the walls is not difficult. 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 quantities 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.

What is an area and what is a rectangle

Area is a geometric quantity that can be used to determine the size of any surface of a geometric figure.

For many centuries, it was customary that the calculation of area was called quadrature. That is, to find out the area of simple geometric figures, it was enough to count the number of unit squares with which the figures were conventionally covered. And a figure that had an area was called squarable.

Therefore, we can summarize that area is a quantity that shows us the size of a part of a plane connected by segments.

A rectangle is a quadrilateral whose angles are all right. That is, a four-sided figure that has four right angles and its opposite sides are equal is called a rectangle.

How to find the area of a rectangle

The easiest way to find the area of a rectangle is to take transparent paper, such as tracing paper or oilcloth, and draw it into equal 1 cm squares, and then attach it to the image of the rectangle. The number of filled squares will be the area in square centimeters. For example, in the figure you can see that the rectangle falls into 12 squares, which means its area is 12 square meters. cm.

But to find the area of large objects, such as an apartment, a more universal method is needed, so a formula was proven to find the area of a rectangle by multiplying its length by its width.

Now let's try to write down the rule for finding the area of a rectangle in the form of a formula. Let's denote the area of our figure by the letter S, the letter a will denote its length, and the letter b will denote its width.

As a result, we get the following formula:

S = a * b.

If we apply this formula to the rectangle drawing above, we will get the same 12 sq. cm, because a = 4 cm, b = 3 cm, and S = 4 * 3 = 12 sq. cm.

If you take two identical figures and superimpose them one on top of the other, they will coincide and will be called equal. Such equal figures will also have equal areas and perimeters.

Why know how to find area

Firstly, if you know how to find the area of a figure, then using its formula you can easily solve any problems in geometry and trigonometry.
Secondly, having learned to find the area of a rectangle, you will first be able to solve simple problems, and over time you will move on to solving more complex ones, and learn to find the area of figures that are inscribed in or near a rectangle.
Thirdly, knowing such a simple formula as S = a * b, you get the opportunity to easily solve any simple everyday problems (for example, find S apartments or houses), and over time you will be able to apply them to solving complex architectural projects.

That is, if we completely simplify the formula for finding the area, it will look like this:

P = L x W,

What P stands for is the required area, D is its length, W is its width, and x is the multiplication sign.

Did you know that the area of any polygon can be conditionally divided into a certain number of square blocks that are located inside this polygon? What is the difference between area and perimeter

Let's use an example to try to understand the difference between perimeter and area. For example, our school is located on an area that is fenced with a fence - the total length of this fence will be the perimeter, and the space that is inside the fence will be the area.

Area units

If the perimeter is one-dimensional and is measured in linear units, which are inches, feet and meters, then S refers to two-dimensional calculations and has its own length and width.

And S is measured in square units, such as:

One square millimeter, where S of the square has a side equal to one millimeter;
A square centimeter has S such a square whose side is equal to one centimeter;
A square decimeter is equal to S of this square with a side of one decimeter;
A square meter has S square, the side of which is one meter;
And finally, a square kilometer has S square, the side of which is one kilometer.

To measure the areas of large areas on the Earth's surface, units such as:

One are or one hundred square meters - if the S square has a side of ten meters;
One hectare is equal to S square, the side of which is one hundred meters.

Tasks and exercises

Now let's look at some examples.

In Figure 62, a figure is drawn that has eight squares and each side of these squares is equal to one centimeter. Therefore, S of such a square will be a square centimeter.

If you write it down, it will look like this:

1 cm2. And S of this figure, consisting of eight squares, will be equal to 8 sq. cm.

If you take any figure and divide it into “p” squares with a side equal to one centimeter, then its area will be equal to:

R cm2.

Let's look at the rectangle shown in Figure 63. This rectangle consists of three stripes, and each such strip is divided into five equal squares with a side of 1 cm.

Let's try to find its area. And so we take five squares, and multiply by three strips and get an area equal to 15 sq.cm:

Consider the following example. Figure 64 shows a rectangle ABCD, divided into two parts by the broken line KLMN. Its first part has an area of 12 cm2, and the second has an area of 9 cm2. Now let's find the area of the entire rectangle:

So, take three and multiply by seven and get 21 sq.cm:

3 7 = 21 sq.cm. In this case, 21 = 12 + 9.

And we come to the conclusion that the area of our entire figure is equal to the sum of the areas of its individual parts.

Let's look at another example. And so in Figure 65 a rectangle is shown, which, using the segment AC, is divided into two equal triangles ABC and ADC

And since we already know that a square is the same rectangle, only having equal sides, then the area of each triangle will be equal to half the area of the entire rectangle.

Let's imagine that the side of the square is equal to a, then:

S = a a = a2.

We conclude that the formula for the area of a square will look like this:

And the entry a2 is called the square of the number a.

And so, if the side of our square is four centimeters, then its area will be:

4 4, that is 4 * 2 = 16 sq.cm.

Questions and tasks

Find the area of the figure, which is divided into sixteen squares, the sides of which are equal to one centimeter.
Remember the rectangle formula and write it down.
What measurements need to be made to find out the area of a rectangle?
Define equal figures.
Can different areas have equal figures? What about the perimeters?
If you know the areas of the individual parts of a figure, how can you find out its total area?
Formulate and write down what the area of the square is.

Historical reference

Did you know that the ancient people in Babylon knew how to calculate the area of a rectangle? The ancient Egyptians also made calculations of various figures, but since they did not know the exact formulas, the calculations had small errors.

In his book “Elements,” the famous ancient Greek mathematician Euclid describes various ways to calculate the areas of different geometric figures.

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 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, 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.

The area of ​​the rectangle is equal. Area of ​​a rectangle