How to find the square of the square along its side and its diagonal?

Today, few people do not know how to find the area of the square. Although no, it was already in the distant yesterday ... That is, at a time when everyone knew how to calculate the area of the square, because today, however it may sound absurd, such questions have constantly appeared on the Internet. It's strange, if not more - scary.

Even in elementary school they teach how to find out the area of a square. But first you should learn to determine the area of the rectangles (and the square is still a rectangle, only with equal sides).

It is suggested to take as a basis a certain square measure of the area - square centimeter or square meter. This measure of the area is a square with a side equal to either one centimeter or one meter. Depending on the size of the area to be measured, it can be a hectare (square kilometer) or ar (a square with a side of 100 meters, in another - "weaving"). These squares are laid down mentally on the measured rectangle.

For the experiment, take a small rectangle with sides, for example, equal to 3 and 5 centimeters. For clarity, junior schoolchildren are encouraged to draw a figure on a sheet in a cage, then dividing the rectangle by parallel straight lines along the length and width, placing them at a distance of two cells. Presumably, two cells in an ordinary school notebook correspond to one centimeter. Thus, it turns out that the rectangle is divided into square centimeters, that is, square centimeters are placed in it - measuring measures of the area.

The next step is to count the squares that fit into the rectangle with a side of one centimeter. You can first recount them in the usual way, pointing to each stick. But then it is necessary to use the multiplication table already learned: five bars were obtained, each of three squares. Multiplying them, we easily get 15 square centimeters. In simple terms, the area of any rectangle is multiplied by its length and width.

Replacing the number 5 by "a", and the number 3 by "b", the children easily derive the formula for finding the area of the rectangle. So, it turns out that S = ax b. But this is the formula for the rectangle. We also need to deduce a rule explaining how to find the area of a square!

Yes, it's very simple! The sides of the square are equal, so you can replace the side "b" in this formula with "a". Then the following expression is output: S = ax a. Multiplication of a number by itself takes the square of this number or a number in the second power.

However, there are other ways to find the square of squares. This, of course, is rather a mathematical task. But they solve certain formulas. For example, you are asked to find out how Find the square of a square Not on the side, but on its diagonal.

To solve this problem, there is little knowledge of primary school. We need the Pythagorean theorem. First, we construct a square, for example, NMOP with diagonal NO = m. We obtain two equal isosceles right triangles with base m.

Applying the above theorem, we find the side of a right triangle. NM squared + MO squared = NO squared. But since NM = MO, we get NM squared + NM squared = NO squared. Hence 2 NM in the square = NO squared. Find NM in a square can be divided by NO squared into two.

But NM is squared - this is exactly the answer to the question of how to find the area of a square! And NO is the diagonal of the square. Hence, we can derive a new formula saying that the area of a square is equal to half of its diagonal raised to the second power.

It is possible to derive the formula for finding the area of a square along the radius of a circle inscribed in it, or described around it. But whatever task we decide to do, the foundation remains forever the rule that we study in elementary school - that by multiplying the two sides of a rectangle, you can find out its area.