Starting from grade 5, students begin to get acquainted with the concept of the areas of different figures. A special role is given to the area of the rectangle, since this figure is one of the easiest to learn.
Area concepts
Any figure has its own area, and the calculation of the area is based on a unit square, that is, from a square with a long side of 1 mm, or 1 cm, 1 dm, and so on. The area of such a figure is equal to $1*1 = 1mm^2$, or $1cm^2$, etc. The area, as a rule, is denoted by the letter - S.
The area shows the size of the part of the plane occupied by the figure outlined by the segments.
A rectangle is a quadrilateral in which all angles are of the same degree measure and equal to 90 degrees, and the opposite sides are parallel and equal in pairs.
Particular attention should be paid to the units of length and width. They must match. If the units do not match, they are converted. As a rule, a large unit is converted to a smaller one, for example, if the length is given in dm, and the width is in cm, then dm is converted to cm, and the result will be $cm^2$.
Rectangle area formula
In order to find the area of a rectangle without a formula, you need to count the number of unit squares into which the figure is divided.
Rice. 1. Rectangle divided into unit squares
The rectangle is divided into 15 squares, that is, its area is 15 cm2. It is worth noting that the figure is 3 squares wide and 5 squares long, so to calculate the number of unit squares, you need to multiply the length by the width. The smaller side of the quadrilateral is the width, the longer the length. Thus, we can derive the formula for the area of a rectangle:
S = a b, where a, b are the width and length of the figure.
For example, if the length of the rectangle is 5 cm and the width is 4 cm, then the area will be 4 * 5 = 20 cm 2.
Calculating the area of a rectangle using its diagonal
In order to calculate the area of a rectangle through the diagonal, you must apply the formula:
$$S = (1\over(2)) ⋅ d^2 ⋅ sin(α)$$
If the task gives the values of the angle between the diagonals, as well as the value of the diagonal itself, then you can calculate the area of the rectangle using the general formula for arbitrary convex quadrilaterals.
A diagonal is a line segment that connects opposite points of a figure. The diagonals of the rectangle are equal, and the intersection point is bisected.
Rice. 2. Rectangle with drawn diagonals
Examples
To consolidate the topic, consider examples of tasks:
No. 1. Find the area of the garden plot, such a shape as in the figure.
Rice. 3. Drawing for the problem
Solution:
In order to subtract the area, it is necessary to divide the figure into two rectangles. One of them will have dimensions of 10 m and 3 m, the other 5 m and 7 m. Separately, we find their areas:
$S_1 =3*10=30 m^2$;
This will be the area of the garden plot $S = 65 m^2$.
No. 2. Subtract the area of the rectangle given its diagonal d=6 cm and the angle between the diagonals α=30 0 .
Solution:
The value of $sin 30 =(1\over(2)) $,
$ S =(1\over(2))⋅ d^2 ⋅ sinα$
$S =(1\over(2)) * 6^2 * (1\over(2)) =9 cm^2$
Thus, $S=9 cm^2$.
The diagonal divides the rectangle into 4 shapes - 4 triangles. In this case, the triangles are pairwise equal. If you draw a diagonal in a rectangle, then it divides the figure into two equal right triangles. Average rating: 4.4. Total ratings received: 214.
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 doesn't 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.
L * H = S to find the area of a rectangle, you need to multiply the width by the length. In other words, it can be expressed like this: the area of a rectangle is equal to the product of the sides.
1. Let's give an example of calculation how to find the area of a rectangle, the sides are equal to known values, for example, width 4 cm, length 8 cm.
How to find the area of a rectangle with sides 4 and 8 cm: The solution is simple! 4 x 8 = 32 cm2. To solve such a simple problem, you need to calculate the product of the sides of the rectangle or simply multiply the width by the length, this will be the area!
2. A special case of a rectangle is a square, this is the case when the sides of the rectangle are equal, in this case, you can find the area of the square using the above formula.
What is the area of the rectangle?
The ability to calculate the area of a rectangle is a basic skill for solving a huge number of everyday or technical problems. This knowledge is applied in almost all areas of life! For example, in cases where areas of any surfaces are needed in construction or real estate. When calculating the areas of land, plots, walls of houses, residential premises ... it is not possible to name a single area of human activity where this knowledge cannot be useful!
If a calculating the area of a rectangle causes you difficulties - just use our calculator! O will immediately bring all the necessary calculations and write the text of the decision with explanations in detail.
Lesson and presentation on the topic: "Perimeter and area of a rectangle"
Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions. All materials are checked by an antivirus program.
Teaching aids and simulators in the online store "Integral" for grade 3
Simulator for grade 3 "Rules and exercises in mathematics"
Electronic textbook for grade 3 "Mathematics in 10 minutes"
What is a rectangle and a square
Rectangle is a quadrilateral with all right angles. So the opposite sides are equal to each other.
Square is a rectangle with equal sides and angles. It is called a regular quadrilateral.
Quadrilaterals, including rectangles and squares, are denoted 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 the 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 the 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 a rectangle ABCD is denoted as P ABCD, where A, B, C, D are the vertices of the rectangle.
Let's write the formula for the perimeter of quadrilateral ABCD:
P ABCD = AB + BC + CD + AD = 2 * AB + 2 * BC = 2 * (AB + BC)
Example.
A rectangle ABCD is given 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 initial data. 
2. Let's write a formula for calculating the perimeter of this rectangle:
P ABCD = 2 * (AB + BC)
P ABCD=2*(5cm+3cm)=2*8cm=16cm
Answer: P ABCD = 16 cm.
The formula for calculating the perimeter of a square
We have a formula for finding the perimeter of a rectangle.P ABCD=2*(AB+BC)
Let's use it to find 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. Determine the perimeter of the square.
Solution.
1. Draw a square ABCD with the original data.
2. Recall the formula for calculating the perimeter of a square:
P ABCD=4*AB
3. Substitute our data into the formula:
P ABCD=4*6cm=24cm
Answer: P ABCD = 24 cm.
Problems for finding 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 CEOM square 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 piece of land is 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 extra material for building a fence.
2. Parents decided to make repairs in the children's room. You need to know the perimeter of the room and its area in order to correctly calculate the number of wallpapers.
Determine the length and width of the room you live in. Determine the perimeter of your room.
What is the area of a rectangle?
Square- This is a numerical characteristic of the figure. The 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 the Latin letter S.
To find 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 AK by the width of KM. Let's write this 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 \u003d AK * KM \u003d 7 cm * 2 cm \u003d 14 cm 2.
Answer: 14 cm 2.
The 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 the square is calculated by multiplying side AB by width BC, but since they are equal, side AB is multiplied by AB.
S ABCO = AB * BC = AB * AB
Example.
Find the area of the 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 a square
1. A rectangle with sides of 20 mm and 60 mm is given. Calculate its area. Write your answer in square centimeters.2. A suburban area was bought with a size of 20 m by 30 m. Determine the area of \u200b\u200bthe summer cottage, write down the answer in square centimeters.
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 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 for calculating the area of a rectangle.
How to find the area of a rectangle on two sides
To find the area of a rectangle, multiply its length by its width: Area = Length × Width. In the case below: Area = AB × BC.
How to find the area of a rectangle given the side and length of the diagonal
In some problems, you need 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, you can determine the second side of the rectangle using the Pythagorean theorem. After that, the problem is reduced to the previous point.
How to find the area of a rectangle by 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 (for example, the width), you can calculate the area of the rectangle using the following formula:
Area \u003d (Perimeter × Width - Width ^ 2) / 2.
Area of a rectangle in terms of the sine of an 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, use the following formula: Area = Diagonal^2 × sin(acute angle between the diagonals)/2.
