How to find the area of a rhombus? To give an answer, you first need to figure out what we consider a rhombus.
Firstly, it is a quadrangle. Secondly, it has all four equal sides. Thirdly, its diagonals at the intersection point are perpendicular. Fourth, these diagonals are divided into equal parts by the intersection point. Fifth, these same diagonals divide the corners of the rhombus into two equal parts. Sixth, in total, two angles that are adjacent to one of the sides make up a developed angle, that is, 180 degrees. And to put it simply, a rhombus is a beveled square.
If you take a square, the sides of which are movably fastened, and it is easy to pull it by two opposite corners, then the square will lose its rectangularity and turn into a rhombus. Therefore, a rhombus with right angles - this is the real square.
The first to introduce the concept of the rhombus Hero and Papp of Alexandria, mathematicians of ancient Greece. The word "rhombus" from Greek can be translated as "tambourine."
To find the area of a rhombus, it is worth considering that a rhombus is a parallelogram. And the area of the parallelogram can be found by multiplying the base, that is, the side, and the height.
To prove this position, perpendiculars should be omitted from the vertices of the upper corners of the rhombus. For example, given the rhombus QWER. Perpendiculars QT and WY are omitted from the vertices of the upper angles Q and W. Moreover, the perpendicular QT will lower to the side RE, and the perpendicular WY will be on the continuation of this side.
Thus, we got a new quadrangle QWYT with parallel sides and right angles, which, based on the foregoing, can be bravely called a rectangle.
The area of this rectangle is found by multiplying the sides and the heights. Now we need to prove that the area of the resulting rectangle in area corresponds to the rhombus given the condition.
Considering the triangles QYR and WET obtained with additional construction, we can say that they are equal in leg and hypotenuse. After all, legs in triangles are drawn perpendiculars, which at the same time are also sides of the resulting rectangle. And hypotenuses are the sides of a rhombus.
The rhombus consists of the sum of the area of the triangle QYR and the trapezoid QYEW. The resulting rectangle consists of the same QYEW trapezoid and the WET triangle, whose area is equal to the area of the QYR triangle. From here the conclusion suggests itself: the value of the area of the rhombus QWER corresponds to the value of the area of the rectangle QWYT.
Now it becomes clear how to find the area of the rhombus on the side and its height: they need to be multiplied.
You can find the area of the rhombus, knowing the angle of the rhombus and the side. It is only necessary to find out what the sine of the angle is, and multiply it by twice the side. You can find the sine using a calculator or from the Bradis table.
Sometimes, when talking about how to find the area of a rhombus, they use the sine of the angle and the radius of the circle inscribed in it, which is necessarily the maximum.
However, most often they calculate the area of a rhombus through the diagonals. From this formula it follows that the area is equal to the semi-product of the diagonals.
To prove this is quite simple by examining two triangles QWE and ERQ, which turned out when one diagonal was drawn in the rhombus. These triangles are equal on three sides or on the base and two adjacent corners.
Having drawn the second diagonal in the rhombus, we get the heights in these triangles, since the diagonals intersect at point X at an angle of 90 degrees. The area of the triangle QWE is equal to the product of QE, which is one diagonal, on WX - half the second diagonal, divided by two.
Now the question of how to find the area of the rhombus, the answer is clear: the resulting expression should be doubled. For the convenience of algebraic reduction of this expression, one diagonal can be denoted by the letter z, and the second by the letter u. We get:
2 (z X 1 / 2u: 2) = z X 1 / 2u, which just comes out - the semi-product of the diagonals.