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 sine 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.
To do it right calculate and set the diagonal of the foundation or formwork foundation - it’s very good to hire specialists. But if you have already seen the “square meter” program several times, heard several times a conversation about how to build, and also a bunch of jokes about construction? - another thing. This gives us “every right” to assume that we ourselves can cope with this simple matter, like the corners and diagonals of the foundation formwork. This is exactly the high opinion of himself that everyone who plans to build a bathhouse with their own hands (Ha-ha!)
I wrote about the beginning of marking and designing the foundation and formwork in the article. At the time of driving in the stakes and installing the outer formwork boards, I already checked the length of the diagonal. Everything converged down to the millimeter. This is the most important condition for obtaining right angles for a bathhouse log. But after the first marking, there were manipulations with installing the bottom of the grillage, installing internal formwork panels, and finishing the formwork of the columns from ground level to the bottom of the future foundation. Of course, I tried very hard not to move anything, and drove the stakes deep.
But as with any construction project, something went wrong. It's not as scary as if I didn't notice it or I didn't know about it. Therefore, before laying the reinforcement, I decided to check the diagonals again. The difference was 2 cm. It’s good that it was discovered before the concrete was poured.
How to display the diagonal of the formwork?
To simplify the construction of the correct formwork, I made the length of the walls absolutely equal. Therefore, the distortion can only be in the form of a diamond. In the figure, the degree of skew of the formwork is deliberately increased for clarity.
To correct the situation, we did this:
This combined movement of one side of the formwork (north side in the picture) was not too difficult since the stakes and the original arrangement of the formwork were in the correct position. Therefore, the diagonal displacement was minimal and efforts to “adjust” the position of the shields did not cause mechanical stress and effort.
The method of setting angles along equal diagonals can only be used if the sides are equal. Diagonal Equalities will be sufficient!
For formwork sides with large size It is possible to apply the golden triangle rule. If such a triangle, according to the Pythagorean theorem, has sides 3, 4, then the hypotenuse is equal to 5 units. Thus, it is enough to measure on the sides of the formwork parts that are multiples of 3 and 4 at the top right angle and then the distance between the control points will be 5 parts! This will guarantee right angles and equality of diagonals!
For proper planning formwork installation I highly recommend using the cast-off method, which allows you to installation work check corners, remove and reinstall foundation perimeter cords.
Before pouring the foundation, do not be too lazy to check the diagonals again. It won't be superfluous! Concrete cannot be fixed easily or quickly. Errors are very expensive and time-consuming to correct. The foundation for a log house has more quality requirements than the foundation for a stone house. Nothing can be leveled with mortar!
Don't forget to remove it before pouring for easy removal!
When solving problems on school math Often you need to determine what the diagonal of a given square is. Despite the apparent complexity, this task is very simple and has several simple solutions. Let's look at them, first introduce some concepts and definitions.
- Square is a quadrilateral with equal sides, all angles of which are right angles, that is, equal to 90 degrees. This figure It is both a rhombus and a rectangle at the same time, therefore it retains all their properties.
- Diagonal of a polygon- this is a segment connecting two of it opposite vertices. In this article we will denote it by the letter d.
- Opposite vertices that do not lie on the same side are called.
- Square root of a number, this is a number that, when multiplied by itself, will give the original. In geometry they are used only positive values square root. In this article we will denote it by the abbreviation rad (from the Latin radical - root).
- We will denote the side of the square by the letter a.
As is clear from the above, a square has only two diagonals. Since a square is a rectangle and retains its properties, they are equal to each other. Let's consider various methods for finding its length.
Calculating the diagonal of a square using a known side
The most in a simple way is diagonal calculation, if the side of the square is known. The well-known Pythagorean theorem applies here for right triangles. Let's write this formula: c^2 = a^2+b^2.
Note that in our case the diagonal of the square is the hypotenuse of the triangle with equal legs. Let's rewrite the formula based on our conditions: d^2 = a^2+a^2. Let's transform, we get: d^2 = 2*a^2. Next step let's extract Square root, it will turn out: d = rad2*a. This is our final formula.
Let's look at the calculation using an example. Let a = 64. Let's substitute our value into the formula. We get d = 64*rad2. This is the answer.
Calculating the diagonal of a square from a known area
Let us be given the area of a square, it is denoted Latin letter S, let's find its diagonal.
We use the properties of a rectangle and write the formula for its area.
S = a*b. Let's rewrite for b = a. We get: s = a^2. From here we find the side: a = radS. So, we managed to express the side through the area. Let us substitute the resulting expression into final formula from the previous part. The formula will look like: d = rad2*a = rad2*radS.
Example: Let's say the area is 32 square meters. Let's substitute this number. We get rad2*rad32 = rad2*4*rad2 = 4*2 = 8 meters.
Calculating the diagonal from a known perimeter
Let us know the perimeter. In the future we will write it with the Latin letter P and find it d. Let's use the properties of a rectangle and write down the formula for its perimeter.
P = two*(a + b). Let's rewrite for b = a. We get: P = two*(a + a) = 2*2a = 4*a. Let us express the side from the last formula. We have: a = P/4. Let's use the fact that: d = rad2*a. Let's express the side through the perimeter. Our formula will take the form d = rad2*P/4.
Example: let the perimeter be 128 meters. Let's do a simple calculation. We have, rad =d2*128/4 = 32*rad2 meters.
Calculation by circumscribed and inscribed circle radius
Another way, which is actually very simple. The radius of the circumscribed circle will be denoted by the Latin letter R, the radius of the inscribed circle will be denoted by the Latin letter r.
First, let's deal with the circumscribed circle. In this situation, its radius is exactly half the diagonal (this is easy to verify using the construction), thus: R = 1/2*d. hence we have: d = two*R. Let us again explain our reasoning with an example. Let R = 45 kilometers. We get d = two*45 = 90 kilometers.
And finally, let's consider a method related to the radius of an inscribed circle. Again, from the construction it is clearly visible that the diameter of the inscribed circle is equal to the side of the square. Thus, its radius is doubled smaller side. Let's write this as a formula: r = 1/2*a. It follows that a = 2*r. Let's use the formula from the first method again and substitute its expression in terms of the radius of the inscribed circle instead of the side. The expression will take the form: d = rad2*a = rad2*2*r.
Let's use an example again. Let r = 98 meters. Then we have, d = rad2*2*98 = 196*rad2.
Conclusion
Thus, in the article we examined five fundamental various methods calculating the diagonal of a square. If, at first glance, the task seemed difficult, then after our reasoning it became obvious that there were no special problems here. Let's summarize all the formulas we received in one table.
- d = rad2*a;
- d = rad2*radS;
- d = rad2*P/4;
- d = 2*R;
- d = rad2*2*r.
I would also like to note that using the first of our formulas it is very easy to construct a segment, equal to root square of the two. To do this, we build a square with side one, its diagonal will be equal to the desired segment.
If we build a rectangle on the resulting diagonal, using it as the length, and take the width equal to one, then we get a segment equal to one more irrational number square root of three.
Video
From the video you will learn how to find the diagonal of a square if its area is known.
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Definition.
Rectangle is a quadrilateral in which two opposite sides are equal and all four angles are equal.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 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 opposite corners equal to 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 opposite sides are not equal to each other (a circle can only be inscribed in special case rectangle - 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: