One of the first formulas that is studied in mathematics is related to how to calculate the area of a rectangle. She is also the most frequently used. Rectangular surfaces surround us everywhere, so you often need to know their area. At least in order to find out if there is enough paint available for painting 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 area of walls, ceiling or floor. They indicate the area of housing.
When it comes to smaller objects, square decimeters, centimeters or millimeters are introduced. The latter are needed if the figure is no larger than a nail.
When measuring the area of a city or country, square kilometers are the most suitable. But there are also units that are used to indicate the size of the area: ar and hectare. The first of them is also 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 to simply multiply both known quantities: length and width. The formula looks like this: S = a * c. Here, the letters a and b indicate the length and width.
Similarly, the area of a 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 checkered paper?
In this situation, you need to rely on the number of cells inside the figure. By their number, it is easy to calculate the area of a rectangle. But this can be done when the sides of the rectangle coincide with the cell lines.
Often there is a position of the rectangle, in which its sides are inclined with respect to the ruled paper. Then the number of cells is difficult to determine, so the calculation of the area of the rectangle is complicated.
You will first need to find out the area of the rectangle, which can be drawn in the cells exactly around the given one. It is simple: multiply the height and width. Then subtract from the resulting value the area of all right triangles. And there are four of them. By the way, they are calculated as half the product of the legs.
The final result will give the value of the area of this rectangle.
What to do 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 you need to calculate its sides in order to use the already familiar formula. First, you need to remember the property of its diagonals. They are equal and divide the intersection point in half. You can see in the drawing that the diagonals divide the rectangle into four isosceles triangles that are pairwise equal to each other.
The equal sides of these triangles are defined as half the diagonal that is known. That is, in each triangle there are two sides and the angle between them, which are given in the task. You can use the cosine theorem.
One side of the rectangle will be calculated by the formula in which 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 of 180 and the known angle.
Now the task of how to calculate the area of the rectangle is reduced to a simple multiplication of the two sides obtained.
What if the perimeter is given in the task?
Usually, the condition also indicates the ratio of length to width. The question of how to calculate the area of a rectangle, in this case, is easier with a concrete example.
Suppose that in the problem the perimeter of a rectangle is 40 cm. It is also known that its length is one and a half times greater than the width. It is necessary to find out its area.
The solution to the problem begins by writing the perimeter formula. It is more convenient to paint it as the sum of the 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 related to the well-known aspect ratio. 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 into 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 an unknown width, the second is the product of the numbers 2 and 1,5 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 remains only to multiply these two numbers to find out the area of the rectangle.
Calculations give the following values: width - 8 cm, length - 12 cm, and area - 96 cm 2 . The last number is the answer to the considered problem.