How to find the volume of a cube in different ways

If you imagine ordinary children's cubes, then you can easily understand how to find the volume of the cube. Taking the volume of one cubic per cubic measure of the volume, for example, per cubic decimeter, we begin to build a large cube from them. Having combined the first square "floor", for example, with the dimensions 4X4, it is necessary to lay out 4 more "floors", so that all the edges of our cube are equal. Equality of all sides of the cube is the basic rule, which proves that it is the cube in front of us.

Find the size of one square face is easy, just multiply the width and length of the base, that is, raise the edge to a square. Since we have several rows - "floors", or rather, they are equal in number to the edge of the cube, then multiply the resulting square by the height of the cube, that is, on its edge. It turns out, in this way, that we raise the edge to the third power, in another way, into the cube. It's so easy, it turns out, to find the volume of the cube!

It is from here and takes its name erection in the third degree - "in the cube". That is, to "cube" it is necessary to multiply the number three times by itself - the expression itself already has in its basis a solution to the problem of finding the cubic volume.

But if the value of the cubic edge, that is, one side of the cube, is unknown, but given the diagonal of one of its faces, how to find the volume of the cube? Can this be done? It turns out, and this is quite computable.

Diagonally, the sides should calculate the side of one face and enter its value into the cube, that is, the third power. In order to be clearer, 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 to the square or to the second power. In the right-angled triangle PMN, the MN side is the hypotenuse, and its square is equal to the sum of the squares that are squared.

But we know that the legs are the sides of the square face of the cube. Hence, the result should be divided into two and find the square root. This result will be equal to the magnitude of the side - the edge of the cube. Now the question of how to calculate the volume of a cube is solved in the simplest way. Only we build the cube side in 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, first you need to find the side of the square - the face of the cube. For this it is sufficient to find the square root of a given area. Then the calculated face value is multiplied by a known area.

Sometimes you just need to know how to find the volume of a cube, But there is not one size, no edge, no square of the side of the cube. However, if this problem has data such as density and mass in the condition, then the report can be calculated by multiplying these values: density and mass. The required volume will be obtained in the work.

And if a person does not have a single dimension, what should he do in this case? In practice, one often uses such a simple technique as immersing a body in a liquid. So how do you find the volume of a cube without a centimeter tape or ruler?

It is necessary to measure a certain amount of liquid in a container, for example, in a saucepan, pouring it to the edges. Then put the container in another container. Having immersed a cube in a liquid, it is necessary to try to collect all overflowed liquid. Then, by measuring it with a beaker or jars (this depends on the size of the cube's volume), one can conclude about the volume of the cube - it will be equal to the amount of liquid that the cube has superseded by its immersion.

Unfortunately, it is difficult or even impossible to measure the volume of cubes of this size in this way. But so you can learn the volume of not only a cube, but objects of any shape.

There are also other possibilities of finding the volume of cubes. For example, for a known length of the diagonal of a cube (not a 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 put the result in a cube and get the desired answer.