What is the diagonal of a cube, and how to find it

What is a cube and what diagonals does it have?

Cube (regular polyhedron or hexahedron)is a three-dimensional shape, each side is a square, which, as we know, all sides are equal. The diagonal of the cube is the segment that passes through the center of the figure and connects the symmetrical vertices. In a regular hexahedron, there are 4 diagonals, 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 cube diagonal

The diagonal of a regular polyhedron can be foundby a very simple formula to remember. D = a√3, where D denotes the diagonal of the cube, and a 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 simple D = 2√3, even it is not necessary to count anything. 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 to find the diagonal of the face of the cube

Diago

The face can also be found by the formula. The diagonals that lie on the faces are only 12 pieces, and they are all equal. 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, two sides of a square and a diagonal form a right-angled triangle. In this trio, the diagonal plays the role of the hypotenuse, and the sides of the square are the legs that are the same length. Let us recall the theorem of Pythagoras, and everything will immediately fall into place. Now the problem: the edge of the hexahedron equals √8 cm, it is necessary to find the diagonal of its face. We paste it into the formula, and we get d = √8 √2 = √16 = 4. Answer: The diagonal of the face of the cube is 4 cm.

If the diagonal of the cube face is known

By the condition of the problem, we are given only the diagonalof the vertex 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, starting from our second formula d = a√2. We obtain 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 the surface area is known

The following algorithm of the solution is based on finding the diagonal over the surface area of ​​the cube. Suppose that it is equal to 72 cm². To begin with, we find the area of ​​one face, and all of them. Hence, 72 must be divided by 6, we get 12 cm². This is the area of ​​one face. To find the edge of a regular polyhedron, it is necessary to recall the formula S = a², then a = √S. We substitute and obtain a = √12 (the edge of the cube). 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 cm².

If the length of the edges of the cube is known

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