If you imagine ordinary children's cubes, then you can easily understand how to find the volume of the cube. Having taken the volume of one cube as a cubic measure of volume, for example, as a cubic decimeter, we begin to build a large cube from them. Having folded the first square “floor”, for example, in 4x4 dimensions, another 4 “floors” should be laid out so that all the edges of our cube are equal. The equality of all sides of the cube is the basic rule that proves that we are faced with a cube.
It’s easy to find the size of one square face, you just have to multiply the width and length of the base, that is, to square the edge. Since we get several rows - "floors", or rather, they get an equal number of cube edges, we multiply the resulting square by the height of the cube, that is, by its edge. It turns out, in this way, that we raise the edge to the third degree, in another way - to the cube. It’s so simple, it turns out, to find the volume of the cube!
It is from here that the raising to the third degree - “into a cube” takes its name. That is, for “erection into a cube” you need to multiply the number three times by itself - the expression itself already has as its basis the solution to the problem of finding the cubic volume.
But if the size of the cubic edge, that is, one side of the cube, is unknown, but the diagonal of one of its faces is given, how to find the volume of the cube? Can this be done? It turns out, and it is quite computable.
On the side diagonal, you should calculate the side of one face and enter its value in a cube, that is, in the third power. To make it clearer, we will draw one of the cubic faces - this will be a square, for example, PMNK, where MN is the diagonal that we know. Using the Pythagorean theorem, we raise the known value of the diagonal in a square or in the second power. In a right-angled triangle PMN, the side MN is a hypotenuse, and its square is equal to the sum of legs squared.
But we know that the legs are the sides of the square face of the cube. So, the result should be divided into two and find the square root. This result will be equal to the size of the side - the edges of the cube. Now the question is how to calculate the volume of a cube, is solved in the simplest way. Just raise the cube side to the third degree - and the result is obvious.
It often happens that in the condition of the problem there is such a value as the area of one of the faces of the cube. In this case, you first need to find the side of the square - the face of the cube. To do this, it is enough to find the square root of a given area. Then, the calculated face value is multiplied by the known area.
Sometimes you just need to know how to find the volume of a cube, but there is not a single size, nor rib, nor square of the side of the cube. However, if this task has data such as density and mass in the condition, then you can calculate the report by multiplying these quantities: density and mass. The desired volume will be obtained in the work.
And if a person does not have a single dimension what to do in this case? In practice, they often use such a simple technique as immersing the body in a liquid. So how to find the volume of a cube without a centimeter tape or ruler?
You need to measure a certain amount of liquid in a container, for example, in a pan, pouring it to the brim. Then you should put the container in another dish. Having immersed the cube in the liquid, you need to try to collect all the liquid that has spilled over the edge. Then, having measured it with a beaker or banks (this depends on the size of the cube), we can draw a conclusion about the volume of the cube - it will be equal to the amount of liquid that the cube displaced by immersion.
Unfortunately, it is quite difficult or even impossible to measure in this way the volume of cubes of considerable size. But in this way you can find out the volume of not only a cube, but objects of any shape.
There are other possibilities for finding the volume of cubes. For example, with the known length of the diagonal of the cube (do not face!). It is known that the formula for the diagonal of a cube is expressed by the product of its edge by the square root of 3. Therefore, divide the diagonal by the square root of 3 and obtain the length of the edge. Then everything is very simple: we raise the result into a cube and get the desired answer.