In a diamond, diagonals are the same

rhombus


Definition: Under a rhombus or one Rhombus one understands a flat square with sides of equal length.

In particular, a diamond is always a convex square. In addition, each diagonal divides the diamond into two congruent isosceles triangles. This means that the two diagonals bisect each other and are perpendicular to each other. Conversely, a diamond can also be characterized by these properties of its diagonals.

So a rhombus is a special one parallelogram with sides of equal length. Therefore the heights in this parallelogram are the same and the rhombus has an inscribed circle. On the other hand, a rhombus only has a circumference if both diagonals are the same length, which is only for that square applies.

Any rhombus thus has two reflection axes, namely its diagonals, and a center of symmetry, the intersection of the two diagonals. There is a 180 turn around this centerO possible.

Except in the special case of the square, in which all four angles are the same, a rhombus has two identical acute and two identical obtuse angles, which are opposite one another. An obtuse and an acute angle complement each other. The shape of a rhombus is clearly determined by one of these angles, for example the smaller angle alpha with. The shape is also completely determined by the ratio of the longer diagonal to the shorter diagonal. The following applies here, with the case precisely describing the squares.

 

Obviously, the following applies to the relationship between alpha and rho

Describes the side length of the rhombus, then according to the Pythagorean theorem, one can express each of these quantities by the other two (or one of the others and).

The area of ​​the rhomb is obviously too

Because of and you get from this too

If you connect the middle of two sides of a diamond to each other, you get a rectangle. Conversely, if you connect the middle of the sides of any rectangle with each other, you get a rhombus. It is said that both squares, i.e. rhombus and rectangle, dual are to each other. Here rho is the ratio of the sides of the rectangle and, and alpha is the (smaller) angle between the two diagonals of the rectangle. Since rectangles and diamonds are dual to one another, they also have the same symmetry mappings.

For the formation of rhombic bodies, i.e. convex bodies whose surfaces consist of rhombuses of the same size, two types of rhombi come into question (if there are more than six surfaces), those with a diagonal ratio of and from, for the ratio of the golden section. This results in approximate values ​​for alpha of 70.5288 ...O in the first or 63.4349 ...O in the second case.

There is another special diamond for, since in this case the diamond consists of two equilateral triangles. The following therefore applies here.