What is a cube, and what diagonals does it have
A cube (regular polyhedron or hexahedron) is a three-dimensional figure, each face is a square, which, as we know, all sides are equal. The diagonal of a cube is a segment that passes through the center of the figure and connects symmetrical vertices. There are 4 diagonals in the correct hexahedron, and they will all 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 on its base. The diagonal of the face of the cube passes through the center of the face and connects the opposite vertices of the square.
The formula by which you can find the diagonal of a cube
The diagonal of a regular polyhedron can be found by a very simple formula that must be remembered. D = a√3, where D is the diagonal of the cube, and is the edge. Let us give an example of a problem where it is necessary to find the diagonal, if it is known that the length of its edge is 2 cm. Here everything is simple D = 2√3, even nothing needs to be considered. In the second example, if the edge of the cube is equal to √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 a cube face
Diago
The nal face can also be found by the formula. Diagonals that lie on the edges, only 12 pieces, and they are all equal to each other. Now 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. To understand where this formula came from is very simple. After all, the two sides of the square and the diagonal form a
right triangle. 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 immediately falls 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 of the face of the cube is known
By the condition of the problem, we are only given the diagonal of the face of the regular polyhedron, which is, say, √2 cm, and we need to find the diagonal of the cube. The formula for solving this problem is a bit 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 value is known, then finding the diagonal of the cube is not difficult: D = 1√3 = √3. That's how we solved our problem.
If surface area is known
The following decision algorithm is based on finding the diagonal over the surface area of the cube. Suppose that it is 72 cm 2 . To begin with, we find the area of one face, and there are 6 in total. Therefore, 72 must be divided by 6, we get 12 cm 2 . This is the area of one face. To find the edge of a regular polyhedron, it is necessary to recall the formula S = a 2 , which means a = √S. Substitute and get a = √12 (cube edge). And if we know this value, then it is not difficult to find the diagonal D = a√3 = √12 √3 = √36 = 6. Answer: the diagonal of the cube is 6 cm 2 .
If the length of the edges of the cube is known
There are cases when in the problem only the length of all the edges of the cube is given. Then it is necessary to divide this value by 12. That is how many sides in a regular polyhedron. For example, if the sum of all the edges is 40, then one side will be 40/12 = 3.333. We insert into our first formula and get the answer!