Definition.

Rectangle is a quadrilateral with two opposite sides are equal and all four angles are the same.

The rectangles differ from each other only in the ratio of the long side to the short side, but all four corners are right, that is, 90 degrees.

The long side of a rectangle is called rectangle length, and the short one - rectangle width.

The sides of a rectangle are also its heights.

Basic properties of a rectangle

A rectangle can be a parallelogram, a square or a rhombus.

1. The opposite sides of the rectangle have the same length, that is, they are equal:

AB = CD, BC = AD

2. Opposite sides of the rectangle are parallel:

3. The adjacent sides of a rectangle are always perpendicular:

AB ┴ BC, BC ┴ CD, CD ┴ AD, AD ┴ AB

4. All four corners of the rectangle are straight:

∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°

5. The sum of the angles of a rectangle is 360 degrees:

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°

6. The diagonals of a rectangle have the same length:

7. The sum of the squares of the diagonal of a rectangle is equal to the sum of the squares of the sides:

2d 2 = 2a 2 + 2b 2

8. Each diagonal of a rectangle divides the rectangle into two identical figures, namely right triangles.

9. The diagonals of the rectangle intersect and are divided in half at the intersection point:

		AO=BO=CO=DO=	d
			2

10. The point of intersection of the diagonals is called the center of the rectangle and is also the center of the circumcircle

11. The diagonal of a rectangle is the diameter of the circumcircle

12. You can always describe a circle around a rectangle, since the sum of opposite angles is 180 degrees:

∠ABC = ∠CDA = 180° ∠BCD = ∠DAB = 180°

13. A circle cannot be inscribed in a rectangle whose length is not equal to its width, since the sums of the opposite sides are not equal to each other (a circle can only be inscribed in a special case of a rectangle - a square).

Sides of a rectangle

Definition.

Rectangle length is the length of the longer pair of its sides. Rectangle width is the length of the shorter pair of its sides.

Formulas for determining the lengths of the sides of a rectangle

1. Formula for the side of a rectangle (length and width of the rectangle) through the diagonal and the other side:

a = √ d 2 - b 2

b = √ d 2 - a 2

2. Formula for the side of a rectangle (length and width of the rectangle) through the area and the other side:

b = dcos	β
	2

Diagonal of a rectangle

Definition.

Diagonal rectangle Any segment connecting two vertices of opposite corners of a rectangle is called.

Formulas for determining the length of the diagonal of a rectangle

1. Formula for the diagonal of a rectangle using two sides of the rectangle (via the Pythagorean theorem):

d = √ a 2 + b 2

2. Formula for the diagonal of a rectangle using the area and any side:

4. Formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle:

d = 2R

5. Formula for the diagonal of a rectangle in terms of the diameter of the circumcircle:

d = D o

6. Formula for the diagonal of a rectangle using the sine of the angle adjacent to the diagonal and the length of the side opposite to this angle:

8. Formula for the diagonal of a rectangle through the sine of the acute angle between the diagonals and the area of ​​the rectangle

d = √2S: sin β

Perimeter of a rectangle

Definition.

Perimeter of a rectangle is the sum of the lengths of all sides of a rectangle.

Formulas for determining the length of the perimeter of a rectangle

1. Formula for the perimeter of a rectangle using two sides of the rectangle:

P = 2a + 2b

P = 2(a + b)

2. Formula for the perimeter of a rectangle using area and any side:

P=	2S + 2a 2	=	2S + 2b 2
	a		b

3. Formula for the perimeter of a rectangle using the diagonal and any side:

P = 2(a + √ d 2 - a 2) = 2(b + √ d 2 - b 2)

4. Formula for the perimeter of a rectangle using the radius of the circumcircle and any side:

P = 2(a + √4R 2 - a 2) = 2(b + √4R 2 - b 2)

5. Formula for the perimeter of a rectangle using the diameter of the circumscribed circle and any side:

P = 2(a + √D o 2 - a 2) = 2(b + √D o 2 - b 2)

Area of ​​a rectangle

Definition.

Area of ​​a rectangle called the space limited by the sides of the rectangle, that is, within the perimeter of the rectangle.

Formulas for determining the area of ​​a rectangle

1. Formula for the area of ​​a rectangle using two sides:

S = a b

2. Formula for the area of ​​a rectangle using the perimeter and any side:

5. Formula for the area of ​​a rectangle using the radius of the circumscribed circle and any side:

S = a √4R 2 - a 2= b √4R 2 - b 2

6. Formula for the area of ​​a rectangle using the diameter of the circumcircle and any side:

S = a √D o 2 - a 2= b √D o 2 - b 2

Circle circumscribed around a rectangle

Definition.

A circle circumscribed around a rectangle is a circle passing through the four vertices of a rectangle, the center of which lies at the intersection of the diagonals of the rectangle.

Formulas for determining the radius of a circle circumscribed around a rectangle

1. Formula for the radius of a circle circumscribed around a rectangle through two sides:

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 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 possible solution.

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%.

Instructions

Length rectangle can be found in several ways. It all depends on the source data.

Option one is perhaps the simplest.

If the width is known rectangle and its area, we use the area formula. It is known that the area rectangle product of width and length rectangle.

Perimeter rectangle it is possible to find by adding the width and length values ​​and multiplying the resulting number by two. We find the unknown side.

We divide the perimeter by two and subtract the width from the resulting figure.

If only the width is known rectangle and the length of the diagonal, you can use the Pythagorean theorem. Divide the rectangle into two equal rectangles.

Next method: the angle between the diagonals is known rectangle and diagonal. Consider the triangle formed rectangle and halves of diagonals. Using the cosine theorem you will find this side rectangle.

Sources:

• find the width of the rectangle
• What is the length of a rectangle if its width is known?

Each of us learned about what a perimeter is in elementary school. Finding the sides of a square with a known perimeter usually does not cause problems even for those who graduated from school a long time ago and managed to forget the mathematics course. However, not everyone can solve a similar problem regarding a rectangle or right triangle without prompting.

Instructions

Suppose that there is a right triangle with sides a, b and c, in which one of the angles is 30 and the other is 60. The figure shows that a = c*sin?, and b = c*cos?. Knowing that the perimeter of any figure, in and triangle, is equal to the sum of all its sides, we obtain:a+b+c=c*sin ?+c*cos+c=pFrom this expression we can find the unknown side c, which is the hypotenuse for the triangle . So what's the angle? = 30, after transformation we get: c*sin ?+c*cos ?+c=c/2+c*sqrt(3)/2+c=p It follows from this that c=2p/Accordingly, a = c*sin ?= p/,b=c*cos ?=p*sqrt(3)/

As mentioned above, the diagonal of a rectangle divides it into two right triangle with angles of 30 and 60 degrees. Since it is equal to p=2(a + b), width a and length b of a rectangle can be found based on the fact that the diagonal is the hypotenuse of right triangles:a = p-2b/2=p/2
b= p-2a/2=p/2These two equations are rectangles. From them, the length and width of this rectangle are calculated, taking into account the resulting angles when drawing its diagonal.

Video on the topic

note

How to find the length of a rectangle if the perimeter and width are known? Subtract twice the width from the perimeter, then we get twice the length. Then we divide it in half to find the length.

Helpful advice

More from primary school Many people remember how to find the perimeter of any geometric figure: it is enough to find out the length of all its sides and find their sum. It is known that in a figure such as a rectangle, the lengths of the sides are equal in pairs. If the width and height of a rectangle are the same length, then it is called a square. Typically, the length of a rectangle is the largest side, and the width is the smallest.

Sources:

• what is the perimeter width in 2019

Tip 3: How to find the area of ​​a triangle and a rectangle

Triangle and rectangle are the two simplest plane geometric figures in Euclidean geometry. Inside the perimeters formed by the sides of these polygons, there is a certain section of the plane, the area of ​​which can be determined in many ways. The choice of method in each specific case will depend on the known parameters of the figures.

Instructions

Use one of the formulas using trigonometric formulas to find the area of ​​a triangle if the values ​​of one or more angles in are known. For example, with a known angle (α) and the lengths of the sides that make it up (B and C), the area (S) can be calculated using the formula S=B*C*sin(α)/2. And with the values ​​of all angles (α, β and γ) and the length of one side in addition (A), you can use the formula S=A²*sin(β)*sin(γ)/(2*sin(α)). If, in addition to all angles, (R) of the circumscribed circle is known, then use the formula S=2*R²*sin(α)*sin(β)*sin(γ).

If the angles are not known, then to find the area of ​​the triangle you can use trigonometric functions. For example, if (H) is drawn from a side that also knows (A), then use the formula S=A*H/2. And if the lengths of each side (A, B and C) are given, then first find the semi-perimeter p=(A+B+C)/2, and then calculate the area of ​​the triangle using the formula S=√(p*(p-A)* (p-B)*(p-C)). If, in addition to (A, B and C), the radius (R) of the circumscribed circle is known, then use the formula S=A*B*C/(4*R).

To find the area of ​​a rectangle, you can also use trigonometric functions - for example, if you know the length of its diagonal (C) and the size of the angle it makes on one of the sides (α). In this case, use the formula S=С²*sin(α)*cos(α). And if the lengths of the diagonals (C) and the size of the angle they make (α) are known, then use the formula S=C²*sin(α)/2.

You can do without trigonometric functions when finding the area of ​​a rectangle if you know the lengths of its perpendicular sides (A and B) - you can use the formula S=A*B. And if the length of the perimeter (P) and one side (A) is given, then use the formula S=A*(P-2*A)/2.

Video on the topic

Division is one of the basic arithmetic operations. It is the opposite of multiplication. As a result of this action, you can find out how many times one of the given numbers is contained in another. In this case, division can replace an infinite number of subtractions of the same number. Problem books regularly contain the task of finding an unknown dividend.

You will need

• - calculator;
• - a sheet of paper and a pencil.

Instructions

Label the unknown dividend as x. Write known data either using given numbers or alphabetic symbols. For example, a task might look like this: x:a=b. Moreover, a and b can be any numbers, both , and . A quotient in the form of an integer means that the division is performed without a remainder. To find the dividend, multiply the quotient by the divisor. The formula will look like this: x=a*b.

If the divisor or quotient is not an integer, remember the features of multiplying fractions and decimals. In the first case, the numerators and denominators are multiplied. If one number is an integer and the other is a simple fraction, the numerator of the second is multiplied by the first. Decimals are multiplied in the same way as integers, but the number of digits to the right of the decimal point is added together, with the trailing zero being included.

Suppose that two sides of a rectangle having one common point(i.e. its length) are given by the coordinates of three points A(X₁,Y₁), B(X₂,Y₂) and C(X₃,Y₃). The fourth point need not be considered - its coordinates do not affect in any way. The length of the projection of side AB onto the abscissa axis will be equal to the difference between the corresponding coordinates of these points (X₂-X₁). The length of the projection onto the ordinate axis is determined similarly: Y₂-Y₁. This means that the length of the side itself, according to the Pythagorean theorem, can be found as the square root

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 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:

• 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 high degree symmetry, then to calculate its perimeter one of the parameters is sufficient. 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 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

The square represents geometric figure, consisting of four sides of equal length and four right angles, each of which is 90°. Determination of area or perimeter quadrilateral, and any one 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.

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 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 right triangles. 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 of the acute angle between them, you should use the following formula: Area = Diagonal^2 × sin(acute angle between the diagonals)/2.