Starting in grade 5, students begin to become familiar with the concept of areas of different shapes. A special role is given to the area of the rectangle, since this figure is one of the easiest to study.
Area Concepts
Any figure has its own area, and the calculation of the area is based on a unit square, that is, on 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 measurement of length and width. They must match. If the units do not match, they are converted. As a rule, they convert a larger unit into 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 takes up 3 squares in width and 5 in length, 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 equal to 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 need to 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 point of intersection is divided in half.
Rice. 2. Rectangle with drawn diagonals
Examples
To reinforce the topic, consider examples of tasks:
No. 1. Find the area of a garden plot of the same shape as in the figure.
Rice. 3. Drawing for the problem
Solution:
In order to subtract the area, you need 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 if given its diagonal d = 6 cm and the angle between the diagonals α = 30 0.
Solution:
Value $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 diagonals divide the rectangle into 4 shapes - 4 triangles. In this case, the triangles are equal in pairs. If you draw a diagonal in a rectangle, it divides the figure into two equal right triangles. Average rating: 4.4. Total ratings received: 214.
Instructions
For example, you know that the length of one of the sides (a) is 7 cm, and perimeter rectangle(P) is equal to 20 cm. Since perimeter of any figure is equal to the sum of the lengths of its sides, and rectangle opposite sides are equal, then its perimeter a will look like this: P = 2 x (a + b), or P = 2a + 2b. From this formula it follows that you can find the length of the second side (b) using a simple operation: b = (P – 2a) : 2. So, in our case, side b will be equal to (20 – 2 x 7) : 2 = 3 cm .
Now, knowing the lengths of both adjacent sides (a and b), you can substitute them into the area formula S = ab. In this case rectangle will be equal to 7x3 = 21. Please note that the units of measurement will no longer be , but square centimeters, since you also multiplied the lengths of the two sides of their units of measurement (centimeters) by each other.
Sources:
- What is the perimeter of a rectangle?
A flat figure consisting of four sides and four right angles. Of all the figures square rectangle have to be calculated more often than others. This and square apartments, and square garden plot, and square table or shelf surfaces. For example, to simply wallpaper a room, they calculate square its rectangular walls.
Instructions
By the way, from rectangle can be easily calculated square. It is enough to complete the rectangular one to rectangle so that the hypotenuse becomes a diagonal rectangle. Then it will be obvious that square such rectangle is equal to the product of the legs of the triangle, and square of the triangle itself, accordingly, is equal to half the product of the legs.
Video on the topic
A special case of a parallelogram - a rectangle - is known only in Euclidean geometry. U rectangle All angles are equal, and each of them separately makes 90 degrees. Based on private properties rectangle, and also from the properties of a parallelogram about the parallelism of opposite sides can be found sides figures along given diagonals and the angle from their intersection. Calculating sides rectangle is based on additional constructions and application of the properties of the resulting figures.
Instructions
Use the letter A to mark the point of intersection of the diagonals. Consider the EFA formed by the constructs. According to property rectangle its diagonals are equal and bisected by the intersection point A. Calculate the values of FA and EA. Since triangle EFA is isosceles and its sides EA and FA are equal to each other and respectively equal to half of the diagonal EG.
Next, calculate the first EF rectangle. This side is the third unknown side of the triangle EFA under consideration. According to the cosine theorem, use the appropriate formula to find the side EF. To do this, substitute the previously obtained values of the sides FA EA and the cosine of the known angle between them α into the cosine formula. Calculate and record the resulting EF value.
Find the other side rectangle F.G. To do this, consider another triangle EFG. It is rectangular, where the hypotenuse EG and leg EF are known. According to the Pythagorean theorem, find the second leg of FG using the appropriate formula.
Refers to the simplest flat geometric figures and is one of the special cases of a parallelogram. A distinctive feature of such a parallelogram is right angles at all four vertices. Limited by parties rectangle square can be calculated in several ways, using the dimensions of its sides, diagonals and angles between them, the radius of the inscribed circle, etc.
Instructions
If the magnitude of the angle (α) that makes up the diagonal is known rectangle on one of its sides, as well as the length (C) of this diagonal, then to calculate the area you can use the definitions of trigonometric in a rectangular. The right triangle here is formed by two sides of the quadrilateral and its diagonal. From the definition of cosine it follows that the length of one of the sides will be equal to the product of the length of the diagonal and the angle, the value is known. From the definition of sine, we can derive the formula for the length of the other side - it is equal to the product of the length of the diagonal and the sine of the same angle. Substitute these identities into the formula from the previous step, and it turns out that to find the area you need to multiply the sine and cosine of a known angle, as well as the length of the diagonal rectangle: S=sin(α)*cos(α)*С².
If, in addition to the diagonal length (C) rectangle If the magnitude of the angle (β) formed by the diagonals is known, then to calculate the area of the figure you can also use one of the trigonometric functions - sine. Square the length of the diagonal and multiply the result by half the sine of the known angle: S=С²*sin(β)/2.
If the (r) of the circle inscribed in the rectangle is known, then to calculate the area, raise this value to the second power and quadruple the result: S=4*r². A quadrilateral into which it is possible will be a square, and the length of its side is equal to the diameter of the inscribed circle, that is, twice the radius. The formula is obtained by substituting the lengths of the sides, expressed in terms of the radius, into the identity from the first step.
If the lengths (P) and one of the sides (A) are known rectangle, then to find the area inside this perimeter, calculate half the product of the side length and the difference between the length of the perimeter and the two lengths of this side: S=A*(P-2*A)/2.
Video on the topic
Not only students in geometry lessons are faced with the task of finding the perimeter or area of a polygon. Sometimes it happens to be solved by an adult. Have you ever had to calculate the required amount of wallpaper for a room? Or maybe you measured the length of your summer cottage in order to enclose it with a fence? Thus, knowledge of the basics of geometry is sometimes indispensable for the implementation of important projects.
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 how the diagonal divided the rectangle into two equal triangles? Therefore, the area of one triangle is 2 times less than the area of a rectangle. So, we need to reduce 12 by 2 times.
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. Mathematics lessons: Methodological recommendations for teachers. 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 papers. 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 for rectangles with an area of 18 cm 2.
4. Create an assignment on the topic of the lesson for your friends.
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 quantities, 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 areas of land, plots, walls of houses, living quarters... it is impossible to name a single area of human activity where this knowledge cannot be useful!
If calculating the area of a rectangle causes you difficulties - just use our calculator! O will instantly provide all the necessary calculations and write the text of the solution with explanations in detail.
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, 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. 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.
