When solving, it is necessary to take into account that solving the problem of finding the area of ​​a rectangle only from the length of its sides it is forbidden.

This is easy to verify. Let the perimeter of the rectangle be equal to 20 cm. This will be true if its sides are 1 and 9, 2 and 8, 3 and 7 cm. All these three rectangles will have the same perimeter, equal to twenty centimeters. (1 + 9) * 2 = 20 is exactly the same as (2 + 8) * 2 = 20 cm.
As you can see, we can select endless number of options the dimensions of the sides of the rectangle, the perimeter of which will be equal to the specified value.

The area of ​​rectangles with a given perimeter of 20 cm, but with different sides, will be different. For the example given - 9, 16 and 21 square centimeters, respectively.
S 1 = 1 * 9 = 9 cm 2
S 2 = 2 * 8 = 16 cm 2
S 3 = 3 * 7 = 21 cm 2
As you can see, there are an infinite number of options for the area of ​​a figure for a given perimeter.

Note for the curious. In the case of a rectangle with a given perimeter, the maximum area will be a square.

Thus, in order to calculate the area of ​​a rectangle from its perimeter, you must know either the ratio of its sides or the length of one of them. The only figure that has an unambiguous dependence of its area on its perimeter is a circle. Only for circle and a solution is possible.

In this lesson:

• Problem 4. Changing the length of the sides while maintaining the area of ​​the rectangle

Problem 1. Find the sides of a rectangle from the area

The perimeter of the rectangle is 32 centimeters, and the sum of the areas of the squares built on each of its sides is 260 square centimeters. Find the sides of the rectangle.
Solution.

2(x+y)=32
According to the conditions of the problem, the sum of the areas of the squares constructed on each of its sides (four squares, respectively) will be equal to
2x 2 +2y 2 =260
x+y=16
x=16-y
2(16-y) 2 +2y 2 =260
2(256-32y+y 2)+2y 2 =260
512-64y+4y 2 -260=0
4y 2 -64y+252=0
D=4096-16x252=64
x 1 =9
x 2 =7
Now let’s take into account that based on the fact that x+y=16 (see above) at x=9, then y=7 and vice versa, if x=7, then y=9
Answer: The sides of the rectangle are 7 and 9 centimeters

Problem 2. Find the sides of a rectangle from the perimeter

The perimeter of the rectangle is 26 cm, and the sum of the areas of the squares built on its two adjacent sides is 89 square meters. cm Find the sides of the rectangle.
Solution.
Let's denote the sides of the rectangle as x and y.
Then the perimeter of the rectangle is:
2(x+y)=26
The sum of the areas of the squares built on each of its sides (there are two squares, respectively, and these are squares of width and height, since the sides are adjacent) will be equal to
x 2 +y 2 =89
We solve the resulting system of equations. From the first equation we deduce that
x+y=13
y=13-y
Now we perform a substitution in the second equation, replacing x with its equivalent.
(13-y) 2 +y 2 =89
169-26y+y 2 +y 2 -89=0
2y 2 -26y+80=0
We solve the resulting quadratic equation.
D=676-640=36
x 1 =5
x 2 =8
Now let's take into account that based on the fact that x+y=13 (see above) at x=5, then y=8 and vice versa, if x=8, then y=5
Answer: 5 and 8 cm

Problem 3. Find the area of ​​a rectangle from the proportion of its sides

Find the area of ​​a rectangle if its perimeter is 26 cm and its sides are proportional as 2 to 3.

Solution.
Let us denote the sides of the rectangle by the proportionality coefficient x.
Hence the length of one side will be equal to 2x, the other - 3x.

Then:
2(2x+3x)=26
2x+3x=13
5x=13
x=13/5
Now, based on the data obtained, we determine the area of ​​the rectangle:
2x*3x=2*13/5*3*13/5=40.56 cm 2

Problem 4. Changing the length of the sides while maintaining the area of ​​the rectangle

The length of the rectangle is increased by 25%. By what percentage should the width be reduced so that its area does not change?

Solution.
The area of ​​the rectangle is
S = ab

In our case, one of the factors increased by 25%, which means a 2 = 1.25a. So the new area of ​​the rectangle should be equal to
S2 = 1.25ab

Thus, in order to return the area of ​​the rectangle to the initial value, then
S2 = S/1.25
S2 = 1.25ab / 1.25

Since the new size a cannot be changed, then
S 2 = (1.25a) b / 1.25

1 / 1,25 = 0,8
Thus, the value of the second side must be reduced by (1 - 0.8) * 100% = 20%

Answer: width should be reduced by 20%.

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 that 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

Useful advice

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

Sources:

• Solution 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 the radius of the incircle of an equilateral triangle is known, then to calculate its perimeter, use the following formula:

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 deduced 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 using known area use the following ratio:
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 quadrilateral, any kind at that, is required not only when solving problems in geometry, 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.

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, not meter and centimeter. The area is recorded Latin letter S. For convenience, let’s denote the length by the Latin letter b, and the width by 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.

Resume

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, they may be able to help. But at least once in our lives we are faced with finding the area of ​​a rectangle.

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 any figure equal to the sum 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 a 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. Distinctive feature of such a parallelogram - 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) that the diagonals form is known, then to calculate the area of ​​the figure you can also use one of 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 extent summer cottage to fence it off? Thus, knowledge of the basics of geometry is sometimes indispensable for the implementation of important projects.

A rectangle is special case quadrangle. 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 equal to 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 to calculate the area of ​​a rectangle.

How to find out the area of ​​a rectangle based on two sides

In order to find the area of ​​a rectangle, you need to multiply its length by its width: Area = Length × Width. In the case given below: Area = AB × BC.

How to find out the area of ​​a rectangle by side and diagonal length

Some problems require you 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 parts right triangle. Therefore, we can determine the second side of the rectangle using the Pythagorean theorem. After this, the task is reduced to the previous point.

How to find out the area of ​​a rectangle by its 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 (such as the width), you can calculate the area of ​​the rectangle using the following formula:
Area = (Perimeter×width – width^2)/2.

Area of ​​a rectangle through the sine of the 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 acute angle between them, you should use the following formula: Area = Diagonal^2 × sin(acute angle between diagonals)/2.