How to find the perimeter of a rectangle on its sides, by its area and one side, by the angle between its diagonal and the side of the rectangle

Often in life, people need to find the perimeter of a rectangle. This problem occurs, for example, in cases where it is necessary to calculate the length of the fence or the number of necessary wallpaper for gluing walls in a room. True, in the latter case, the perimeter is only an intermediate link in the solution of a practical problem. But, nevertheless, in this case, people also need to know how to find the perimeter of a rectangle.

To begin with, I would like to determine what a perimeter is. Perimeter, in fact, the boundary of a geometric figure or the total length of its boundary. Now let us explain the meaning of the concept of a rectangle. A parallelogram with right angles should be referred to rectangles. Actually, the main distinguishing feature is precisely its right angles, which must be in this geometric figure four.

Thus, in order to find the total length of the boundary of the rectangle, it is necessary to add the lengths of all its sides. As we have already explained, the parallel sides in the rectangle are equal, hence, for the sake of understanding, it should be understood that the perimeter of the rectangle is equal to twice the sum of its two sides.

For clarity, we will designate equal sides in the rectangle with the letters "a" and "b" of the Latin alphabet, respectively. Thus, it turns out that P (perimeter of the rectangle) = a + b + a + b. This equation can be transformed into the following formula: P = 2X (a + b).

However, in life there are often situations where the length of only one side and any other parts in the rectangle is known, or outside it. Let's consider some variants.

For example, we need to calculate what the perimeter of a rectangle is equal to, provided that the length of one of the sides in the rectangle is unknown, but its area is known. It is necessary, using the formula for calculating the area of a rectangle, which is equal to the product of its sides, to calculate the length of its second side. This is easy to do by dividing a known area into a known side. Knowing both sides of the rectangle, you can easily calculate its perimeter.

This option is suitable for calculating the required amount of material for the site fence, when its area is specified in the documentation. It is only necessary to measure one of the sides of the plot. But sometimes you need to know how to find the perimeter of a rectangle, if one of the sides of the rectangle and its diagonal is known.

Naturally, the first step in the calculation is to find the length of the second side of the rectangle. It can be calculated by the Pythagorean theorem, which states that the hypotenuse of a right-angled triangle, squared, includes the sum of the squares of both sides. Therefore, to calculate, we need to build the length of the diagonal and the length of the known side into a square, then find the difference between them, and already from this difference we need to extract the square root.

The resulting square root will be the length of the unknown side. And since you can find the perimeter of a rectangle by adding the known lengths of the sides and doubling them, then everyone can easily cope with this process.

Mathematics lessons also consider the question of how to find the perimeter of a rectangle by its diagonal and one sharp corner formed by a diagonal and one of the sides of a rectangle. Here we have a classic example of using the value of a sine in the calculation. From the school program, everyone knows that the sine of the angle of a rectangular triangle is equal to the ratio of the adjacent leg and the hypotenuse. From this follows the formula: sin X = cathet: hypotenuse (diagonals of the rectangle).

Sinus can be easily recognized by the Bradys table, the known value of the diagonal - hypotenuse is substituted in the formula, and one of the sides of the rectangle is easily computed. Now the next step is to find the second side of the rectangle. Here, the above-considered variant comes into force using the Pythagoras theorem. We squared the known diagonal and subtract from the obtained number the square of the found side. From the answer, we extract the square root. From the now known sides, you can calculate the perimeter by adding their lengths and doubling them.

Naturally, this is not an exhaustive version of the examples, in fact there are many more, but the ones described above are most often encountered.

Thus, we can conclude that without knowing the lengths of two parallel sides, the perimeter of the rectangle is almost impossible to determine. However, applying the arsenal of geometric theorems and axioms, one can always calculate the perimeter of a rectangle, adding all its sides.