The area of ​​a rectangle may not sound arrogant, but it is an important concept. In everyday life we ​​constantly encounter it. Find out the size of fields, vegetable gardens, calculate the amount of paint needed to whitewash the ceiling, how much wallpaper will be needed for pasting

money and more.

Geometric figure

First, let's talk about the rectangle. This is a figure on a plane that has four right angles and its opposite sides are equal. Its sides are usually called length and width. They are measured in millimeters, centimeters, decimeters, meters, etc. Now we will answer the question: “How to find the area of ​​a rectangle?” To do this, you need to multiply the length by the width.

Area=length*width

But one more caveat: length and width must be expressed in the same units of measurement, that is, meter and meter, and not meter and centimeter. The area is written with the Latin letter S. For convenience, let’s denote the length with the Latin letter b, and the width with the Latin letter a, as shown in the figure. From this we conclude that the unit of area is mm 2, cm 2, m 2, etc.

Let's look at a specific example of how to find the area of ​​a rectangle. Length b=10 units. Width a=6 units. Solution: S=a*b, S=10 units*6 units, S=60 units 2. Task. How to find out the area of ​​a rectangle if the length is 2 times the width and is 18 m? Solution: if b=18 m, then a=b/2, a=9 m. How to find the area of ​​a rectangle if both sides are known? That's right, substitute it into the formula. S=a*b, S=18*9, S=162 m 2. Answer: 162 m2. Task. How many rolls of wallpaper do you need to buy for a room if its dimensions are: length 5.5 m, width 3.5, and height 3 m? Dimensions of a roll of wallpaper: length 10 m, width 50 cm. Solution: make a drawing of the room.

The areas of opposite sides are equal. Let's calculate the area of ​​a wall with dimensions of 5.5 m and 3 m. S wall 1 = 5.5 * 3,

S wall 1 = 16.5 m 2. Therefore, the opposite wall has an area of ​​16.5 m2. Let's find the area of ​​the next two walls. Their sides, respectively, are 3.5 m and 3 m. S wall 2 = 3.5 * 3, S wall 2 = 10.5 m 2. This means that the opposite side is also equal to 10.5 m2. Let's add up all the results. 16.5+16.5+10.5+10.5=54 m2. How to calculate the area of ​​a rectangle if the sides are expressed in different units of measurement. Previously, we calculated areas in m2, then in this case we will use meters. Then the width of the wallpaper roll will be equal to 0.5 m. S roll = 10 * 0.5, S roll = 5 m 2. Now we’ll find out how many rolls are needed to cover a room. 54:5=10.8 (rolls). Since they are measured in whole numbers, you need to buy 11 rolls of wallpaper. Answer: 11 rolls of wallpaper. Task. How to calculate the area of ​​a rectangle if it is known that the width is 3 cm shorter than the length, and the sum of the sides of the rectangle is 14 cm? Solution: let the length be x cm, then the width is (x-3) cm. x+(x-3)+x+(x-3)=14, 4x-6=14, 4x=20, x=5 cm - length rectangle, 5-3=2 cm - width of the rectangle, S=5*2, S=10 cm 2 Answer: 10 cm 2.

Summary

Having looked at the examples, I hope it has become clear how to find the area of ​​a rectangle. Let me remind you that the units of measurement for length and width must match, otherwise you will get an incorrect result. To avoid mistakes, read the task carefully. Sometimes a side can be expressed through the other side, don't be afraid. Please refer to our solved problems, it is quite possible that they can help. But at least once in our lives we are faced with finding the area of ​​a rectangle.

So, first, let's look at the formulas for finding area and perimeter:

1) S = a * b = 56 cm2;

2) P = 2a + 2b = 30 cm.

After all, we know that a rectangle has two identical sides.

Thus, we need to solve a system of two equations:

From this we see that one side is 7 and the other is 8.

Knowing the formulas for the perimeter of a rectangle and its area, the sides are sought in the form of solving a system of two equations. First, we express the value of one side through the other and, for example, the area. It looks like this: A = S / B = 56 / B

Then we substitute this expression for the letter A in the equation for the perimeter:

P=2(56/V + V)=30

We get that 56/B+B=15

In this equation, you don’t even need to solve it - anyone familiar with the multiplication table can immediately see that 56 is the product of 7 and 8, and since the sum of these numbers is just 15, then they are the values ​​​​of the sides of the rectangle we need.

You can try to solve this problem by creating a system of equations.

The perimeter of the rectangle is: p=2a+2b;

The area of ​​the rectangle is: s=a*b;

Since we know the perimeter and area, we immediately substitute the numbers:

Express b in terms of a in the second equation:

And substitute 56/a instead of b in the first equation:

Multiply both sides by a:

We get a quadratic equation:

Finding the roots of this quadratic equation:

(15(15-4*1*56))/2*1 = (15(225-224))/2 = (151)/2 = (151)/2

It turns out that the roots of this equation are:

a1=(15+1)/2=16/2=8;

a2=(15-1)/2=14/2=7;

It turns out that we have 2 possible options for rectangles.

Let's remember what we expressed: b=56/a;

From here we find possible b:

b1=56/a1=56/8=7;

b2=56/a2=56/7=8;

As it turns out, these two different rectangles are one and the same; you can simply achieve a perimeter of 30 with an area of ​​56:

If a=7 and b=8.

Or vice versa: a=8 and b=7.

That is, in essence, we have the same rectangle, it’s just that in one version the vertical side is larger than the horizontal, and in the other, on the contrary, the horizontal is larger than the vertical.

Answer: one side is 7 centimeters, and the other is 8 centimeters.

• Let's remember school geometry:

  The perimeter of a rectangle is the sum of the lengths of all sides, and the area of ​​a rectangle is the product of its two adjacent sides (length times width).

  In this case, we know both the Area and Perimeter of the rectangle. They are 56 cm^2 and 30 cm, respectively.

  So, the solution:

  S - area = a x b;

  P - perimeter = a + b + a + b = 2a + 2b;

  30 = 2 (a + b);

  Let's make a substitution:

  56 = (15 - b) x b;

  56 = 15 b - b^2;

  b^2 - 15b + 56 = 0.

  We got a quadratic equation, solving which we get: b1 = 8, b2 = 7.

  We find the other side of the rectangle:

  a1 = 15 - 8 = 7;

  a2 = 15 - 7 = 8.

  Answer: The sides of the rectangle are 8 and 7 cm or 7 and 8 cm.

  If the perimeter of a rectangle is P = 30 cm and its area is S = 56 cm, then its sides will be equal:

  a - one side, b - the other side of the rectangle.

  Having solved this system, we come to the conclusion that side a will be equal to 7 cm, and side b will be equal to 8 cm.

  a = 7 cm b = 8 cm.

• Given: S = 56 cm

  P = 30 cm

  Sides=?

  Solution:

  Let the sides of the rectangle be a and b.

  Then: area S = a * b, perimeter P=2*(a + b),

  We get a system of equations:

  (a*b=56 ? (ab=56

  (2(a+b)=30, (a+b=15, expressing b through a we get a quadratic equation:

  b=15-a, a^2 -15a +56 =0 , solving which we get:

  b1=8, b2=7. That is, the sides of the rectangle: a=7,b=8, or vice versa: a=8,b=7.

To solve the problem, you need to create a system of equations and solve it

we get a quadratic equation that can be easily solved if we substitute the values ​​of perimeter and area into it

The discriminant is 1 and the equation has two roots 7 and 8, therefore one of the sides equal to 7 cm, the other 8 cm or vice versa.

I specifically wrote out the discriminant here because it is very easy to navigate

if in the condition of the problem of finding the sides of a rectangle, the value of the perimeter and area are specified so that this discriminant more than zero, then we have rectangle;

if discriminant equal to zero- then we have square(P=30, S=56.25, square with side 7.5);

if discriminant less than zero, then like this rectangle does not exist(P=20, S=56 - no solution)

Perimeter 30, area 56. Let's call the sides of the rectangle a and c. Then we can create the following equations:

Let's denote one side by the letter X, the other by the letter Y.

The area of ​​a rectangle is calculated by multiplying the lengths of the sides, so we can formulate the first equation:

The perimeter is the sum of the lengths of the sides, therefore the second equation is:

We obtain a system of two equations.

Using the first equation, select X: X=56:Y, substitute this into the second equation:

2*56:Y+2Y=30 From here it’s easy to find the value of Y: Y=7, then X=8.

I found another solution:

It is known that the perimeter of a rectangle is 30 and the area is 56, then:

perimeter = 2*(length + width) or 2L + 2W

area= length * width or L * W

2L + 2W = 30 (divide both parts by 2)

L * (15 - L) = 56

To be honest, I didn’t quite understand the solution, but I think anyone who hasn’t completely forgotten mathematics will figure it out.

Side A=7, side B=8

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.

4a, where a is the side of a square or rhombus. Then the length sides equal to one fourth of the perimeter: a = p/4.

This problem can also be easily solved for a triangle. He has three of the same length sides, so the perimeter p of an equilateral triangle is 3a. Then the side of the equilateral triangle is a = p/3.

For the remaining figures you will need additional data. For example, you can find sides, knowing its perimeter and area. Suppose the length of the two opposite sides of the rectangle is a, and the length of the other two sides is b. Then the perimeter p of the rectangle is equal to 2(a+b), and the area s is equal to ab. We get a system with two unknowns:
p = 2(a+b)
s = ab. Express from the first equation a: a = p/2 - b. Substitute into the second and find b: s = pb/2 - b². The discriminant of this equation is D = p²/4 - 4s. Then b = (p/2±D^1/2)/2. Discard the root that is less than zero and substitute in for sides a.

Sources:

• Find the sides of a rectangle

If you know the value of a, then you can say that you have solved the quadratic equation, because its roots will be found very easily.

You will need

• -discriminant formula for a quadratic equation;
• -knowledge of multiplication tables

Instructions

Video on the topic

Helpful advice

The discriminant of a quadratic equation can be positive, negative, or equal to 0.

Sources:

• Solving Quadratic Equations
• discriminant even

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.

Tip 4: How to find the perimeter of an equilateral triangle

An equilateral triangle, along with a square, is perhaps the simplest and most symmetrical figure in planimetry. Of course, all relations that are valid for an ordinary triangle are also true for an equilateral triangle. However, for a regular triangle, all formulas become much simpler.

You will need

• calculator, ruler

Instructions

To measure the length of one of its sides and multiply the measurement by three. This can be written as follows:

Prt = Ds * 3,

Prt – perimeter of the triangle,
Ds is the length of any of its sides.

The perimeter of the triangle will be in the same dimensions as the length of its side.

Since an equilateral triangle has a high degree of symmetry, one of the parameters is sufficient to calculate its perimeter. For example, area, height, inscribed or circumscribed circle.

If you know the radius of the incircle of an equilateral triangle, then use the following formula to calculate its perimeter:

Prt = 6 * √3 * r,

where: r is the radius of the inscribed circle.
This rule follows from the fact that the radius of the incircle of an equilateral triangle is expressed in terms of the length of its side by the following relation:
r = √3/6 * Ds.

To calculate the perimeter in terms of the circumradius, use the formula:

Prt = 3 * √3 * R,

where: R is the radius of the circumscribed circle.
This is easily derived from the fact that the circumradius of a regular triangle is expressed through the length of its side by the following relation: R = √3/3 * Ds.

To calculate the perimeter of an equilateral triangle through a known area, use the following relationship:
Srt = Dst² * √3 / 4,
where: Sрт – area of ​​an equilateral triangle.
From here we can deduce: Dst² = 4 * Sрт / √3, therefore: Dst = 2 * √(Sрт / √3).
Substituting this ratio into the perimeter formula through the length of the side of an equilateral triangle, we obtain:

Prt = 3 * Dst = 3 * 2 * √(Srt / √3) = 6 * √Sst / √(√3) = 6√Sst / 3^¼.

Video on the topic

A square is a geometric figure consisting of four sides of equal length and four right angles, each of which is 90°. Determination of area or perimeter a quadrilateral, any kind at that, is required not only when solving geometry problems, but also in everyday life. These skills can become useful, for example, during repairs when calculating the required amount of materials - coverings for floors, walls or ceilings, as well as for laying out lawns and beds, etc.

4. Formula for the radius of a circle, which is described around a rectangle through the diagonal of a square:

5. Formula for the radius of a circle, which is described around a rectangle through the diameter of the circle (described):

6. Formula for the radius of a circle, which is described around a rectangle through the sine of the angle that is adjacent to the diagonal, and the length of the side opposite to this angle:

7. Formula for the radius of a circle, which is described around a rectangle through the cosine of the angle that is adjacent to the diagonal, and the length of the side of this angle:

8. Formula for the radius of a circle, which is described around a rectangle through the sine of the acute angle between the diagonals and the area of ​​the rectangle:

The angle between the side and the diagonal of a rectangle.

Formulas for determining the angle between the side and the diagonal of a rectangle:

1. Formula for determining the angle between the side and the diagonal of a rectangle through the diagonal and side:

2. Formula for determining the angle between the side and the diagonal of a rectangle through the angle between the diagonals:

The angle between the diagonals of a rectangle.

Formulas for determining the angle between the diagonals of a rectangle:

1. Formula for determining the angle between the diagonals of a rectangle through the angle between the side and the diagonal:

β = 2α

2. Formula for determining the angle between the diagonals of a rectangle through area and diagonal.