﻿ Geometric cubes. What is a cube diagonal, and how to find it

# Geometric cubes. What is a cube diagonal, and how to find it

Or a hexahedron) is a three-dimensional figure, each face is a square in which, as we know, all sides are equal. The diagonal of the cube is a segment that passes through the center of the figure and connects symmetric vertices. In a regular hexahedron there are 4 diagonals, and all of them will be equal. It is very important not to confuse the diagonal of the figure itself with the diagonal of its face or square, which lies at its base. The diagonal face of the cube passes through the center of the face and connects the opposite vertices of the square.

Formula for finding the cube diagonal

The diagonal of a regular polyhedron can be found using a very simple formula that needs to be remembered. D = a√3, where D is the diagonal of the cube, and is the edge. We give an example of a problem where it is necessary to find a diagonal, if it is known that the length of its edge is 2 cm. Here everything is just D = 2√3, even nothing needs to be considered. In the second example, let the edge of the cube be √3 cm, then we get D = √3√3 = √9 = 3. Answer: D is 3 cm.

The formula by which you can find the diagonal of the cube face

Diago You can also find a face by the formula. The diagonals that lie on the edges are only 12 pieces, and they are all equal. Now we remember d = a√2, where d is the diagonal of the square, and is also the edge of the cube or the side of the square. Understanding where this formula came from is very simple. After all, the two sides of the square and the diagonal form. In this trio, the diagonal plays the role of the hypotenuse, and the sides of the square are the legs, which have the same length. Recall the Pythagorean theorem, and everything will immediately fall into place. Now the task: the edge of the hexahedron is √8 cm, it is necessary to find the diagonal of its face. We insert into the formula, and we get d = √8 √2 = √16 = 4. Answer: the diagonal of the cube face is 4 cm.

If the diagonal face of the cube is known

By the condition of the problem, we are given only the diagonal of the face of a regular polyhedron, which is, say, √2 cm, and we need to find the diagonal of the cube. The formula for solving this problem is slightly more complicated than the previous one. If we know d, then we can find the edge of the cube, based on our second formula d = a√2. We get a = d / √2 = √2 / √2 = 1cm (this is our edge). And if this quantity is known, then it is easy to find the cube diagonal: D = 1√3 = √3. That's how we solved our problem.

If surface area is known

## The following solution algorithm is based on finding the diagonal by suppose that it is equal to 72 cm 2. To begin with, we will find the area of ​​one face, and there are six of them altogether. So, 72 must be divided by 6, we get 12 cm 2 This is the area of ​​one facet. To find the edge of a regular polyhedron, it is necessary to recall the formula S = a 2, which means a = √S. Substitute and we get a = √12 (edge ​​of the cube). And if we know this value, then the diagonal is not difficult to find D = a√3 = √12 √3 = √36 = 6. The answer: the cube diagonal is 6 cm 2.

If the length of the cube edges is known

There are cases when the problem is given only the length of all edges of the cube. Then it is necessary to divide this value by 12. It is the number of sides in the correct polyhedron. For example, if the sum of all edges is 40, then one side will be equal to 40/12 = 3.333. We insert into our first formula and get the answer!

In which you need to find the edge of the cube. This is the definition of the length of a cube edge by the area of ​​the face of the cube, by the volume of the cube, by the diagonal of the face of the cube and by the diagonal of the cube. Consider all four options for such tasks. (The remaining tasks, as a rule, are variations of the above or tasks in trigonometry, which are very indirectly related to the issue under consideration)

If you know the area of ​​the face of the cube, then find the edge of the cube is very simple. Since the face of the cube is a square with a side equal to the edge of the cube, its area is equal to the square of the edge of the cube. Therefore, the length of the edge of the cube is equal to the square root of the area of ​​its face, that is:

and - the length of the edge of the cube,

S is the area of ​​the cube face.

Finding the face of a cube in its volume is even easier. Given that the volume of the cube is equal to the cube (of the third degree) of the length of the edge of the cube, we obtain that the length of the edge of the cube is equal to the root of the cubic (third degree) of its volume, ie:

and - the length of the edge of the cube,

V is the volume of the cube.

Finding the length of a cube edge along known diagonal lengths is a little more difficult. Denote by:

and - the length of the edge of the cube;

b - the length of the diagonal of the face of the cube;

c - the length of the cube diagonal.

As can be seen from the figure, the diagonal of the face and the edges of the cube form a rectangular equilateral triangle. Therefore, by the Pythagorean theorem:

From here we find:

(to find the edge of the cube you need to extract Square root from half the square of the diagonal face).

To find the edge of the cube along its diagonal, we use the pattern again. The diagonal of the cube (c), the diagonal of the face (b), and the edge of the cube (a) form a right triangle. So, according to the Pythagorean theorem:

We use the above relationship between a and b and substitute in the formula

b ^ 2 = a ^ 2 + a ^ 2. We get:

a ^ 2 + a ^ 2 + a ^ 2 = c ^ 2, whence we find:

3 * a ^ 2 = c ^ 2, therefore:

A cube is a rectangular parallelepiped, all edges of which are equal. Therefore, the general formula for the volume of a rectangular parallelepiped and the formula for its surface area in the case of a cube are simplified. Also, the volume of the cube and its surface area can be found, knowing the volume of the ball inscribed in it, or the ball described around it.

You will need

• the length of the side of the cube, the radius of the inscribed and described ball

Instruction

The volume of a rectangular parallelepiped is: V = abc - where a, b, c are its dimensions. Therefore, the volume of the cube is equal to V = a * a * a = a ^ 3, where a is the length of the side of the cube . The surface area of ​​the cube is equal to the sum of the areas of all its faces. The cube has six faces, so its surface area is S = 6 * (a ^ 2).

Let the ball fit into the cube. Obviously, the diameter of this ball will be equal to the side of the cube . Substituting the length of the diameter in the expression for the volume instead of the length of the cube edge and using that the diameter is equal to twice the radius, we get then V = d * d * d = 2r * 2r * 2r = 8 * (r ^ 3), where d is the diameter of the inscribed circle and r is the radius of the inscribed circle. The surface area of ​​the cube will then be S = 6 * (d ^ 2) = 24 * (r ^ 2).

Let the ball be described around a cube . Then its diameter will coincide with the diagonal of the cube . The diagonal of the cube passes through the center of the cube and connects its two opposite points.
Consider first one of the faces of the cube . The edges of this facet are the legs of a right triangle, in which the diagonal of face d will be a hypotenuse. Then, by the Pythagorean theorem, we obtain: d = sqrt ((a ^ 2) + (a ^ 2)) = sqrt (2) * a.

Then consider the triangle in which the hypotenuse is the diagonal of the cube , and the diagonal of the face d and one of the edges of the cube a is its legs. Similarly, by the Pythagorean theorem, we get: D = sqrt ((d ^ 2) + (a ^ 2)) = sqrt (2 * (a ^ 2) + (a ^ 2)) = a * sqrt (3).
So, according to the derived formula, the diagonal of the cube is D = a * sqrt (3). Hence, a = D / sqrt (3) = 2R / sqrt (3). Therefore, V = 8 * (R ^ 3) / (3 * sqrt (3)), where R is the radius of the described ball. The surface area of ​​the cube is S = 6 * ((D / sqrt (3)) ^ 2) = 6 * (D ^ 2) / 3 = 2 * (D ^ 2) = 8 * (R ^ 2).

Often there are tasks in which you need to find the edge of a cube, often this should be done on the basis of information about its volume, facet area or its diagonal. There are several options for defining a cube edge.

In that case, if the area of ​​the cube is known, then the edge can be easily determined. The face of the cube is a square with a side equal to the edge of the cube. Accordingly, its area is equal to the square edge of the cube. You should use the formula: a = √S, where a is the length of the edge of the cube, and S is the area of ​​the face of the cube. Finding a cube edge by its volume is an even simpler task. It is necessary to take into account that the volume of the cube equals cube (in the third degree) the length of the edge of the cube. It turns out that the length of the edge is equal to the cube root of its volume. That is, we get the following formula: a = √V, where a is the length of the edge of the cube, and V is the volume of the cube. Diagonally, you can also find the edge of the cube. Accordingly, we need: a - the length of the edge of the cube, b - the length of the diagonal of the face of the cube, c - the length of the diagonal of the cube. By the Pythagorean theorem, we get: a ^ 2 + a ^ 2 = b ^ 2, and from here you can easily derive the following formula: a = √ (b ^ 2/2), which extracts the edge of the cube. Once again, using the Pythagorean theorem (a ^ 2 + a ^ 2 = b ^ 2), we can get the following relationship: a ^ 2 + a ^ 2 + a ^ 2 = c ^ 2, from which we derive: 3 * a ^ 2 = c ^ 2, therefore, the edge of the cube can be obtained as follows: a = √ (c ^ 2/3). # Расчет высокопрочных болтов на растяжение

Особенности расчета на прочность элементов, ослабленных отверстиями под высокопрочные болты:
При статической нагрузке, если ослабление менее 15 °/о, расчет ведется по площади брутто А, а если ослабление больше 15 %—по условной площади Лусл = 1,18 Ап.

# Монтажные стыки

Монтажные стыки делают при невозможности транспортирования элементов в целом виде.
Монтажные стыки для удобства сборки устраивают универсальными: все прокатные элементы балки соединяют в одном сечении.

# Проверка прочности

Проверка прочности сечения на опоре балки по касательным напряжениям:
Балочной клеткой называется система перекрестных балок, предназначенная для опирания настила при устройстве перекрытия над какой-либо площадью.