How to calculate the area of a rectangle: practical tips

One of the first formulas that is studied in mathematics is related to how to calculate the area of a rectangle. It is also the most frequently used. Rectangular surfaces surround us everywhere, so it is often necessary to know their areas. At least in order to find out if there is enough paint available for painting the floors.

What area units exist?

If we talk about the one that is accepted as international, it will be a square meter. It is convenient to use when calculating the areas of walls, ceiling or floor. They indicate the area of housing.

When it comes to smaller objects, then enter square decimeters, centimeters or millimeters. The latter are needed if the figure is not bigger than the nail.

When measuring the area of a city or country, the most suitable are square kilometers. But there are also units that are used to indicate the size of the area: ar and hectare. The first of them is called weaving.

What if the sides of the rectangle are given?

This is the easiest way to calculate the area of a rectangle. It is enough simply to multiply both known values: length and width. The formula looks like this: S = a * a. Here the letters a and b denote the length and width.

Similarly, the square of the square is calculated , which is a special case of a rectangle. Since all sides are equal, the product becomes the square of the letter a .

What if the figure is depicted on plaid paper?

In this situation, you must rely on the number of cells inside the figure. By their number it can be simple to calculate the area of a rectangle. But this can be done when the sides of the rectangle coincide with the lines of the cells.

Often there is such a position of the rectangle, in which its sides are inclined with respect to the paper lining. Then the number of cells is difficult to determine, so the calculation of the area of the rectangle becomes more complicated.

It will be necessary to first learn the area of the rectangle, which can be drawn by cells exactly around this. It's simple: multiply the height and width. Then subtract from the resulting value the area of all rectangular triangles. And there are four of them. By the way, they are counted as half the work of the legs.

The final result will give the value of the area of this rectangle.

How to proceed if the parties are unknown, but given its diagonal and the angle between the diagonals?

Before finding the area of a rectangle, in this situation it is necessary to calculate its sides in order to use the already familiar formula. First, we need to recall the property of its diagonals. They are equal and divide by the point of intersection in half. You can see in the drawing that the diagonals divide the rectangle into four isosceles triangles, which are pairwise equal to each other.

The equal sides of these triangles are defined as half the diagonal, which is known. That is, in each triangle there are two sides and an angle between them, which are given in the problem. We can use the cosine theorem.

One side of the rectangle will be calculated by the formula in which the equal sides of the triangle and the cosine of the given angle appear. To calculate the second, the cosine value will have to be taken from an angle equal to the difference 180 and the known angle.

Now the problem of how to calculate the area of a rectangle is reduced to a simple multiplication of the two sides obtained.

What if there is a perimeter in the problem?

Usually the condition also specifies the ratio of length and width. The question of how to calculate the area of a rectangle, in this case, is easier on a concrete example.

Let us assume that in the problem the perimeter of a certain rectangle is 40 cm. It is also known that its length is one and a half times the width. It is necessary to find out its area.

The solution of the problem begins with writing down the perimeter formula. It is more convenient to write it as a sum of length and width, each of which is multiplied by two separately. This will be the first equation in the system to be solved.

The second is due to the well-known relationship of the sides. The first side, that is, the length, is equal to the product of the second (width) and the number 1.5. This equality must be substituted in the formula for the perimeter.

It turns out that it is equal to the sum of two monomials. The first is the product of 2 and of unknown width, the second is the product of numbers 2 and 1.5 and of the same width. In this equation, only one unknown is the width. You need to count it, and then use the second equality to calculate the length. It only remains to multiply these two numbers to find out the area of the rectangle.

Calculations give such values: width - 8 cm, length - 12 cm, and area - 96 cm ² . The last number is the answer to the problem considered.