Which side of the triangle is equal to the diagonal of the rectangle. Rectangle. Formulas and properties of the rectangle

The task of finding the diagonal of the rectangle can be formulated by three different ways. Consider more each of them. Ways depend on the known data, so how to find a diagonal of a rectangle?

If two sides are known

In the case when two sides of the rectangle A and B are known, to find the diagonal, it is necessary to use the Pythagora theorem: A 2 + B 2 \u003d C 2, here a and b - the cathets of the rectangular triangle, C - the hypotenuse of the rectangular triangle. When a diagonal is scored in a rectangle, it is divided into two rectangular triangles. Two sides of this rectangular triangle we are known (a and b). That is, to find the diagonal of the rectangle, the formula is needed as follows: C \u003d √ (A 2 + B 2), here C - the length of the diagonal of the rectangle.

On the well-known side and corner, between the side and diagonal

Let it be known to the side of the rectangle A and the angle that it forms with the diagonal of the rectangle α. To begin with, recall the cosine formula: COS α \u003d A / C, here C is a rectangle diagonal. How to calculate the rectangle diagonal from this formula: C \u003d A / COS α.

According to the well-known side, the corner between the adjacent side of the rectangle and the diagonal.

Since the rectangle diagonal divides the rectangle itself into two rectangular triangles, it is logical to refer to the definition of sinus. Sinus is the ratio of the category lying against this angle, to hypotenuse.sin α \u003d b / c. From here, we derive the formula for finding a diagonal of a rectangle, which is also a hypotenneus of a rectangular triangle: C \u003d b / sin α.

Now you are pamped in this matter. You can please the geometry teacher tomorrow!

Content:

Diagonal is a segment that connects two opposite vertices of the rectangle. In a rectangle, two equal diagonals. If the side of the rectangle is known, the diagonal can be found on the Pythagora theorem, because the diagonal divides the rectangle into two rectangular triangles. If the parties are not given, other values \u200b\u200bare known, for example, an area and perimeter or side attitude, you can find the sides of the rectangle, and then by the Pythagora theorem calculate the diagonal.

Steps

1 on the sides

1. 1 Write down the Pythagore's theorem. Formula: a 2 + b 2 \u003d C 2
2. 2 In the formula, substitute the values \u200b\u200bof the parties. They are given in the task or need to be measured. The parties are substituted instead of a 3
   • In our example:
     4 2 + 3 2 \u003d C 2 4

     2 in area and perimeter

     1. 1 formula: S \u003d L W (in the figure instead of s used designation A.)
     2. 2 This value is substituted instead of s 3 Rewrite the formula to separate W 4 Record the formula to calculate the perimeter of the rectangle. Formula: P \u003d 2 (W + L)
     3. 5 In the formula, substitute the value of the perimeter of the rectangle. This value is substituted instead of P 6 Divide both sides of the equation by 2. You will receive the sum of the sides of the rectangle, namely W + L 7 In the formula, substitute the expression to calculate W 8 Get rid of the fraraty. To do this, both parts of the equation multiply on L 9 Eclay equation to 0. To do this, from both sides of the equation, deduct a member from the first order variable.
        • In our example:
          12 L \u003d 35 + L 2 10 Arrange the members of the equation. The first member will be a member from a second order variable, then a member from a first-order variable, and then a free member. At the same time, do not forget about the signs ("plus" and "minus"), which are facing members. Note that the equation is recorded as a square equation.
          • In our example 0 \u003d 35 + l 2 - 12 l 11
            • In our example, equation 0 \u003d L 2 - 12 L + 35 12 Find L 13. Write down the Pythagore's theorem. Formula: a 2 + b 2 \u003d C 2
              • Take advantage of the Pythagora theorem, because each rectangle diagonal divides it into two equal rectangular triangles. And the side of the rectangle is the triangle katets, and the diagonal of the rectangle - the hypotenus of the triangle.
            • 14 These values \u200b\u200bare substituted instead of 15 Take the length and width into the square, and then fold the results obtained. Remember that when the number is erected into a square, it is multiplied by itself.
              • In our example:
                5 2 + 7 2 \u003d C 2 16 Remove square root From both sides of the equation. Use the calculator to quickly remove the square root. You can also use the online calculator. You will find C.

                3 on the area and the attitude of the parties

                1. 1 Record the equation characterizing the side of the parties. Separate L 2. Record the formula to calculate the rectangle area. Formula: S \u003d L W (in the figure instead of s used designation A.)
                   • This method is applicable and in the case when the value of the perimeter of the rectangle is known, but then you need to use the formula to calculate the perimeter, and not the area. Formula for calculating the perimeter of the rectangle: p \u003d 2 (W + L)
                2. 3 In the formula, substitute the value of the rectangle area. This value is substituted instead of s 4 In the formula, substitute an expression characterizing the side of the parties. In the case of a rectangle, you can substitute an expression for calculating L 5 Write down quadratic equation. To do this, expose brackets and equate the equation to zero.
                   • In our example:
                     35 \u003d w (W + 2) 6 Spread the square equation for multipliers. To obtain detailed instructionsRead.
                     • In our example, equation 0 \u003d W 2 - 12 W + 35 7 Find W 8. Submold the value of the width value (or length) into the equation characterizing the side of the parties. So you can find the other side of the rectangle.
                       • For example, if you calculated that the width of the rectangle is 5 cm, and the side ratio is set by the equation L \u003d W + 2 9 Write down the Pythagore's theorem. Formula: a 2 + b 2 \u003d C 2
                         • Take advantage of the Pythagora theorem, because each rectangle diagonal divides it into two equal rectangular triangles. And the side of the rectangle is the triangle katets, and the diagonal of the rectangle - the hypotenus of the triangle.
                       • 10 In the formula, substitute the lengths and widths. These values \u200b\u200bare substituted instead of 11 Take the length and width into the square, and then fold the results obtained. Remember that when the number is erected into a square, it is multiplied by itself.
                         • In our example:
                           5 2 + 7 2 \u003d C 2 12 Remove the square root from both sides of the equation. Use the calculator to quickly remove the square root. You can also use the online calculator. You will find C (DisplayStyle C), that is, a triangle hypotenus, which means the diagonal of the rectangle.
                           • In our example:
                             74 \u003d C 2 (DisplayStyle 74 \u003d C ^ (2))
                             74 \u003d C 2 (DisplayStyle (SQRT (74) \u003d (SQRT (C ^ (2))))
                             8, 6024 \u003d C (DisplayStyle 8,6024 \u003d C)
                             Thus, a rectangle diagonal, in which 2 cm length is larger than the width and the area of \u200b\u200bwhich is 35 cm 2, is approximately 8.6 cm.

Square - Samea simple figure in geometry. It is from her, a rectangle and square begin to study this subject. The ability to solve the task with a square will help you to master a more complex material. This article will tell you how to find the diagonal of the square.

The solution of geometric tasks is interesting to solve them in several ways. Every way is interesting in its own way. No exception and diagonal of the square, which can be found direct and indirect paths.

How to find a square diagonal - formula

There is a pretty simple formula for finding a diagonal of a square. It looks like this: a√2. A - Square side. Recall that all sides of the square are equal. Thus, if you know the size of the one hand, you know the size of the other three sides. To find out the diagonal of the square, it is necessary to multiply its side to the root of two.

Example 1: Find the diagonal of the square, if it is known that his side is 5.

Decision: Substituting the value in the aforementioned formula, it is not difficult to guess that the diagonal will be equal to 5√2.

Example 2: Find the side of the square, if it is known that its diagonal is 5√2.

Decision: Diagonal is denoted by a small Latin letter d. D \u003d A√2. Therefore, to find the side of knowing diagonal, it is necessary to divide the diagonal to the root of two. Having done this action, we learn the side of the square, which, in this case, is 5.

How to find a square diagonal through a rectangular triangle

If you have a diagonal in the square, it is easy to notice that two rectangular triangles are formed. Recall that a rectangular triangle has one angle necessarily straight. It consists of two cathets (the side at an angle of 90 degrees) and hypotenuses (the opposite of the 90-degrads corner of the side). The square of the hypotenuse is equal to the sum of the squares of the cathets. In this case, hypotenuse is the diagonal of our square. Since kartets are the sides of the square, the formula will have the following form: D² \u003d A² + a² \u003d 2a². It follows that d \u003d √2a² \u003d a√2.

Example 3: Find the diagonal of the square if his side is 3.

Decision:

1. We fold the squares of the parties, we get 18.
2. We consider the root of 18 and get 3√2.

Despite the fact that the last method is longer and ultimately we go out on the formula from the first example, it is necessary to know it. In essence, this method is proof of the formula of the square diagonal. It is this proof that can come to the exam or the Olympiad. Good learn her, because she can help you at the aforementioned events.

Online calculator

Despite the fact that it is not difficult to solve such tasks, some students can forget the formula. For such cases there is online calculatorwhich allows you to find the correct answer based on what is given in the task. To use this service, follow the link.

1. Scroll down the page down and you will find the subtitle "Find a diagonal of a square, knowing the side.
2. Below this subtitle will be given the formula, looking at which you do not need a calculator.
3. But still, if you are not sure, enter the value of the length of the square in the field, and then on the "Calculate" button.
4. Calculator for 1 second will give you the correct answer.

Now, knowing several ways to solve the task of this topic, you will not flip the book on mathematics in search of the desired formula, but simply use the online calculator or examples that are given above.

When solving tasks in school mathematics, it is often necessary to determine what the diagonal of the specified square is equal. With seeming some complexity, this task is very simple and has several uncomplicated ways solutions. Consider them, first introduce some concepts and definitions.

1. Square - This is a quadricle with equal parties, all the angles of which are direct, that is, 90 degrees are equal. This figure is simultaneously both rhombus, and a rectangle, so retains all of their properties.
2. Diagonal polygon - This is a segment connecting two opposite vertices. In the article it will be denoted by the letter D.
3. Opposite They are called vertices that are not lying on one side.
4. Square rootThis is such a number that will give the original when multiplying itself. In geometry used only positive meanings Square root. In the article, we will be denoted by the reduction of RAD (from the Latin Radical - root).
5. Side the square will be denoted by the letter a.

As it is clear from the foregoing, the square has only two diagonals. Since the square is a rectangle and retains its properties, they are equal to each other. Consider various methods for finding its length.

Calculation of the diagonal of the square on the well-known side

Most. simple way is an diagonal calculationIf a square side is known. There is a widely known Pythagoreo theorem for rectangular triangles. We write this formula: C ^ 2 \u003d a ^ 2 + b ^ 2.

Note that in our case the diagonal of the square is the hypotenus of a triangle with equal customs. We rewrite the formula based on our conditions: D ^ 2 \u003d a ^ 2 + a ^ 2. We transform, we obtain: d ^ 2 \u003d 2 * a ^ 2. The next step removed the square root, will turn out: d \u003d RAD2 * A. This is our final formula.

Consider the calculation on the example. Let A \u003d 64. We will substitute our value in the formula. We obtain D \u003d 64 * RAD2. This is the answer.

Calculation of the diagonal of the square on the famous area

Let us be given the square of the square, it is indicated by the Latin letter S, we will find it diagonal.

We use the properties of the rectangle and install the formula of its area.

S \u003d A * b. Refrigerate for B \u003d a. We obtain: s \u003d a ^ 2. From here you find the side: a \u003d rads. So, we managed to express the side through the square. We substitute the resulting expression into the final formula from the previous part. The formula will take the form: d \u003d Rad2 * a \u003d rAD2 * Rads..

Example: Suppose the area is 32 square meters. Substitute this number. We obtain RAD2 * RAD32 \u003d RAD2 * 4 * RAD2 \u003d 4 * 2 \u003d 8 meters.

Calculation of the diagonal at a well-known perimeter

Let us know the perimeter. In the future, we will record the Latin letter P, find it d. We use the properties of the rectangle and install the formula of its perimeter.

P \u003d two * (A + B). Refrigerate for B \u003d a. We will succeed: p \u003d two * (a + a) \u003d 2 * 2a \u003d 4 * a. Express the latest formula. We have: a \u003d p / 4. We use the fact that: d \u003d Rad2 * a. Express the side after the perimeter. Our formula will take the formD \u003d RAD2 * P / 4.

Example: Let the perimeter are 128 meters. Let's do a simple calculation. We have, RAD \u003d D2 * 128/4 \u003d 32 * Rad2 meters.

Calculation by radius described and inscribed circle

Another waywhich is very simple in self. We will denote the radius of the circle described by the Latin letter R, the radius of the inscribed circle will be denoted by the Latin letter R.

First you will deal with the circumference described. In this situation, its radius is exactly half a diagonal (it is not difficult to verify the use of construction), thus: r \u003d 1/2 * d. From here we have: d \u003d two * r. We again explain our reasoning on the example. Let R \u003d 45 kilometers. We get, d \u003d two * 45 \u003d 90 kilometers.

And finally, we consider the method associated with the radius of the inscribed circle. Again, it is clearly clearly seen that the diameter of the inscribed circle is equal to the side of the square. Thus, its radius is twice less side. We write it as a formula: r \u003d 1/2 * a. From here it follows, a \u003d 2 * r. We will use the formula again from the first method, we substitute instead of its expression through the radius of the inscribed circle. The expression will take the form: d \u003d Rad2 * a \u003d rAD2 * 2 * R.

Once again we will use the help of the example. Let R \u003d 98 meters. Then we have, d \u003d Rad2 * 2 * 98 \u003d 196 * Rad2.

Conclusion

So we looked at the article five fundamentally various methods Calculations of the diagonal of the square. If, at first glance, the task seemed difficult, then after the reasoning we held, it became obvious that there were no special problems here. We will minimize all the formulas we got into one table.

1. d \u003d Rad2 * a;
2. d \u003d Rad2 * Rads;
3. d \u003d Rad2 * p / 4;
4. d \u003d 2 * r;
5. d \u003d Rad2 * 2 * R.

I want to note yetWith the first of our formulas, it is very easy to build a segment equal to the root square of two. To do this, we build a square with a side of the unit, its diagonal and will be equal to the desired segment.

If we construct a rectangle on the diagonally received, using it as a length, and take the width to one one, then we get a cut equal to one another irrational number Square root of three.

Video

From the video you will learn how to find a square diagonal if its area is known.

Didn't get the answer to your question? Offer authors the topic.

4. The formula of the circle radius, which is described near the rectangle through the diagonal of the square:

5. The formula of the circle radius, which is described near the rectangle through the circle diameter (described):

6. The formula of the circle radius, which is described near the rectangle through the sine of the angle, which is adjacent to the diagonal, and the length of the side of the opposite corner:

7. The formula of the circle radius, which is described near the rectangle through the cosine of the angle, which is adjacent to the diagonal, and the side length of this angle:

8. The formula of the circle radius, which is described near the rectangle through the sine of an acute angle between the diagonals and the area of \u200b\u200bthe rectangle:

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

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

1. The formula for determining the angle between the side and the diagonal of the rectangle through the diagonal and side:

2. The formula for determining the angle between the side and the diagonal of the rectangle through the angle between the diagonals:

The angle between the diagonals of the rectangle.

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

1. The formula for determining the angle between the diagonals of the rectangle through the angle between the side and the diagonal:

β \u003d 2α

2. The formula for determining the angle between the diagonals of the rectangle through the area and diagonal.