How to find a parallelogram area if the side and height are known? How to find the area of \u200b\u200bthe parallelogram if its diagonals are known or the sides and angle?

In this article one of the mathematical topics will be disclosed. You will learn how to find the area of \u200b\u200bthe parallelogram. This subject is taught in eighth grade. Those who have not dealt with her will be useful for this article.

Content

At school, it happens that the teacher explains the lesson, but the children do not understand. Therefore, it turns out that the child does not learn not only one topic, but those that go further. Especially in geometry. After all, many evidence are derived on the basis of rules and previous theorems. Then we learn how to find the area of \u200b\u200bthe parallelogram. But initially, in order to find out the area, you should know the definition of what parallelograms are. This figure is a quadrangle with parallel sides and equal opposite angles. Now let's find the area of \u200b\u200bthe figure with different methods.

How to find the area of \u200b\u200bthe parallelogram - the properties of the figure

So, the parallelogram looks as follows:

Even the ancient Greek scientist of mathematics Euclid described several properties of this figure in the book “Beginning”. More precisely, two characteristics of the parallelogram:

the figure can also be compared with a rectangle, because everything is, on the contrary, its lying sides are parallel, equal, also intersect at the angles of 90 °.
also, the rule applies to the square, rhombus, only in the corners.

IMPORTANT: Before proceeding with proof, we will decide on the term - the area. The area is the size of the figure itself, or rather the plane occupied by it, which is limited to the sides of this figure.

It is not without reason that these properties are described above, thanks to them it will be easier to find out how to calculate S is the area of \u200b\u200bthe figure.

There are several basic formulas to calculate S - Parallelogram area:

When given: the height and length of the parallelogram
When given: the length of one side of the figure, the angles of the figure
When given: the dimensions of both diagonals, one of the angles of their intersection.

Now about each of these methods in more detail.

Calculation of the area of \u200b\u200bthe parallelogram, if the parties are known, height

To calculate the size S of the figure (parallelogram area), you should know all its properties. These rules have already been considered above. So, the first formula is finding the area of \u200b\u200bthe figure on the side and height. Let VN - height, and AB is a side. Height is carried out at the base at an angle of 90º.

The evidence of this axiom is provided above. It can be seen from it that S \u003d A • H. By the way, the area is measured in square units.

S \u003d AB • VN, to start with the conclusion of the theorem, the triangles formed as a result of the heights to the same base should be considered. They will be equal to each other. Well, then and then the area of \u200b\u200bthe rectangle of the formed will be equal to the area of \u200b\u200bthe parallelogram. And earlier it was proved that in the S rectangle \u003d A • H. That is why parallelograms will have the same formula for calculating the area.

Calculation of the area of \u200b\u200bthe parallelogram by diagonals

You can find the area of \u200b\u200bthe parallelogram with various methods. And this option is common. In order to calculate S, you should know the size of the angle and the length of the diagonals of the parallelogram. This axiom is also important in geometry, knowing it, you can easily solve problems in control and independent work.

For evidence, two equal triangles should be considered, which turned out to be a parallelogram into two parts.

On three sides. So the corners in these triangles are equal, see the picture above. And the area of \u200b\u200bthe triangle is equal to half the work of the side a to height h. And the height in these triangles is the diagonal of the parallelogram. From here it turns out that S parallelogram is equal to the area of \u200b\u200bthese two triangles or 1/2 sin α to the work of diagonals.

S \u003d 1/2 • Sin α • D1 • D2

Which was required to find.

Calculation of the area of \u200b\u200bthe parallelogram, if the parties are known, angle

If you know what the lengths of both sides are equal to, the corner, you can find S parallelogram. The area of \u200b\u200bthe parallelogram in this case is equal:

S \u003d B • A • Sinown.

In order to prove this axiom, it is enough by the formulas to find the height of the figure and substitute the data found in the well -known formula of the parallelogram.

According to the rules of geometry, if you consider the triangles, then the sin of the angle will be equal to the ratio of the opposite H - the leg towards the hypotenuse. But the cattle, this is the height of the figure. So it turns out: