Today, few people do not know how to find the area of a square. Although no, it was already far back yesterday ... That is, at a time when everyone knew how to calculate the square area, because today, no matter how absurd it sounds, such questions constantly began to appear on the Internet. This is strange, to say the least - scary.
Even in elementary school they teach how to find out the area of a square. But first you must learn to determine the area of the rectangles (and the square is still a rectangle, only with equal sides).
It is proposed to take as a basis a certain square measure of area measurement - a square centimeter or square meter. This measure of 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 (square with a side of 100 meters, in other words - “weaving”). These squares are laid mentally on the measured rectangle.
For experience, you should take a small rectangle with sides, for example, equal to 3 and 5 centimeters. For clarity, younger students are invited to draw a figure on a sheet in a cage, then dividing the rectangle with parallel lines in length and width, placing them at a distance of two cells. Presumably, two boxes in a regular 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 - measures for measuring the area.
The next step is to count the squares that fit into the rectangle with a side of one centimeter. You can first count them in the usual way, pointing to each stick. But then you must definitely use the already-learned multiplication table: you get five columns, each with three squares. Multiplying them, we easily get 15 square centimeters. In simple terms, the area of any rectangle is found by multiplying its length and width.
Replacing the number 5 with “a” and the number 3 with “b”, children easily derive a formula for finding the area of a rectangle. So, it turns out that S = ax b. But this is the formula for the rectangle. We need to derive a rule explaining how to find the area of a square!
Yes, very easy! The sides of the square are equal, which means that you can replace side “b” in this formula with “a”. Then the following expression is derived: S = ax a. Multiplying a number by itself gets the square of that number, or a second degree number.
However, there are other ways to find the square area. These, of course, are more likely mathematical problems. But when solving them, certain formulas are derived. For example, asked to learn how find the square area not on the side, but on its diagonal.
To solve such a problem, there is already little knowledge of elementary school. We need the Pythagorean theorem. First, construct a square, for example, NMOP with a diagonal of NO = m. We get 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 squared = NO squared. Find NM squared by dividing NO squared by two.
But NM squared is the answer to the question of how to find the square area! And NO is the diagonal of the square. So, we can derive a new formula, which says that the area of the square is equal to half its diagonal, raised to the second degree.
We can derive a formula for finding the area of a square along the radius of either a circle inscribed in it or a circle circumscribed around it. But no matter what task we decide, the rule that we study in elementary school will always remain the foundation - that by multiplying the two sides of a rectangle, you can find out its area.