Such an amazing and familiar square.It is symmetrical about its center and axes drawn along the diagonals and through the centers of the sides. And to search for the area of a square or its volume does not amount to much difficulty. Especially if the length of his side is known.
The first two properties are related to the definition.All sides of the figure are equal to each other. After all, the square is the right quadrilateral. And he necessarily all sides are equal and the angles have the same value, namely - 90 degrees. This is the second property.
The third is related to the length of the diagonals. They are also equal to each other. And they intersect at right angles and at the points of the middle.
First about the designation. For the length of the side, it is customary to choose the letter "a". Then the square of the square is calculated by the formula: S = a2.
It is easily obtained from the one known forrectangle. In it, the length and width are multiplied. For a square, these two elements are equal. Therefore, the square of this one quantity appears in the formula.
It is the hypotenuse in the triangle, the legswhich are the sides of the figure. Therefore, we can use the formula of the Pythagorean theorem and derive an equality in which the side is expressed through the diagonal.
Carrying out such simple transformations, we obtain that the square of the square through the diagonal is calculated by the following formula:
S = d2 / 2. Here the letter d denotes the diagonal of the square.
In this situation, it is necessary to express the sideThrough the perimeter and substitute it in the area formula. Since there are four sides of the figure, the perimeter will have to be divided by 4. This will be the value of the side, which can then be substituted into the initial one and the area of the square.
The general formula looks like this: S = (P / 4)2.
No. 1. There is a square. The sum of its two sides is 12 cm. Calculate the area of the square and its perimeter.
Decision. Since the sum of the two sides is given, you need to know the length of one. Since they are the same, the known number must be simply divided into two. That is, the side of this figure is 6 cm.
Then its perimeter and area can be easily calculated from the above formulas. The first is 24 cm, and the second is 36 cm2.
Answer. The perimeter of the square is 24 cm, and its area is 36 cm2.
2. Find out the area of the square with a perimeter of 32 mm.
Decision. It is enough to substitute the perimeter value into the above formula. Although you can first know the side of the square, and then its area.
In both cases, the actions will first go to division, and then the exponentiation. Simple calculations lead to the fact that the area of the presented square is 64 mm2.
Answer. The required area is 64 mm2.
The side of the square is 4 dm. Dimensions of the rectangle: 2 and 6 dm. Which of the two figures has more area? How much?
Decision. Let the side of the square be denoted by the letter a1, then the length and width of the rectangle a2 and in2. To determine the area of a square, the value a1 is to be squared, and the rectangle multiplied by a2 and in2 . It is not difficult.
It turns out that the square of the square is 16 dm2, and the rectangle - 12 dm2. Obviously, the first figure is larger than the second one.This is despite the fact that they are equal, that is, they have the same perimeter. To check, you can count the perimeters. At the square, the side should be multiplied by 4, it turns out to be 16 dm. At the rectangle, fold the sides and multiply by 2. There will be the same number.
In the task it is still necessary to answer, on how many areas differ. For this, a smaller number is subtracted from a larger number. The difference is 4 dm2.
Answer. Areas are equal to 16 dm2 and 12 dm2. At the square it is more by 4 dm2.
Condition.A rectangle is constructed on the isotope of an isosceles right triangle. To its hypotenuse height is built on which another square is built. Prove that the area of the first is twice as large as the second.
Decision. We introduce the notation. Let the cathete be equal to a, and the height to the hypotenuse, x. Area of the first square - S1, the second - S2.
The square of the square constructed on the leg is easy to calculate. It turns out to be equal to a2. With the second value, everything is not so simple.
First you need to know the length of the hypotenuse. For this, the formula of the Pythagorean theorem is useful. Simple transformations lead to the following expression: a√2.
Since the height in an isosceles triangle,drawn to the bottom, is also a median and a height, then it divides a large triangle into two equal isosceles right triangles. Therefore, the height is half the hypotenuse. That is, x = (a√2) / 2. Hence it is easy to find the area S2. It turns out to be equal to a2/ 2.
Obviously, the recorded values differ exactly by a factor of two. And the second is a number of times smaller. Q.E.D.
It is made from a square. It is necessary to cut it into various shapes according to certain rules. The total of the pieces must be 7.
The rules assume that during the game all the resulting details will be used. Of these, you need to make other geometric shapes. For example, a rectangle, a trapezoid or a parallelogram.
But it's even more interesting when silhouettes of animals or objects are obtained from pieces. And it turns out that the area of all derived figures is equal to that of the initial square.
