The rhombus (from the ancient Greek ῥόμβος and from the Latin rombus “tambourine”) is a parallelogram, which is characterized by the presence of identical sides in length. In the case when the angles are 90 degrees (or a right angle), such a geometric figure is called a square. Rhombus - a geometric figure, a kind of quadrangles. It can be either a square or a parallelogram.
Origin of the term
Let's talk a little about the history of this figure, which will help to reveal a little for yourself the mysterious secrets of the ancient world. The word that is familiar to us, often found in school literature, “rhombus”, originates from the ancient Greek word “tambourine”. In ancient Greece, these musical instruments were produced in the form of a rhombus or square (in contrast to modern devices). Surely you noticed that the card suit - tambourine - has a rhombic shape. The formation of this suit dates back to the days when round tambourines were not used in everyday life. Therefore, the rhombus is the oldest historical figure that was invented by mankind long before the appearance of the wheel.
For the first time such a word as “rhombus” was used by such famous personalities as Heron and Pope of Alexandria.
Rhombus properties
- Since the sides of the rhombus are opposite to each other and are pairwise parallel, the rhombus is undoubtedly a parallelogram (AB || CD, AD || BC).
- Rhombic diagonals have a right-angle intersection (AC ⊥ BD), which means they are perpendicular. Therefore, the intersection divides the diagonals in half.
- The bisectors of the rhombic angles are the diagonals of the rhombus (∠DCA = ∠BCA, ∠ABD = ∠CBD, etc.).
- From the identity of the parallelograms, it follows that the sum of all the squares of the diagonals of the rhombus is the number of squares of the side, which is multiplied by 4.
Signs of a rhombus
In those cases, a rhombus is a parallelogram when it meets the following conditions:
- All sides of the parallelogram are equal.
- Diagonals of a rhombus cross a right angle, that is, they are perpendicular to each other (AC⊥BD). This proves the rule of three sides (the sides are equal and are at an angle of 90 degrees).
- The diagonals of a parallelogram divide the angles equally, since the sides are equal.
Diamond Square
The area of the rhombus can be calculated using several formulas (depending on the material provided in the task). Next, read about what is the area of the rhombus.
- The area of the rhombus is equal to the number, which is half the product of all its diagonals.
- Since the rhombus is a kind of parallelogram, the area of the rhombus (S) is the number of the product of the side of the parallelogram by its height (h).
- In addition, the area of the rhombus can be calculated by the formula, which is the product of the squared side of the rhombus by the sine of the angle. The sine of the angle — alpha — is the angle between the sides of the original diamond.
- A formula that is a product of the doubled angle alpha and the radius of the inscribed circle (r) is considered to be completely acceptable for the right decision.
You can calculate and prove these formulas based on the Pythagorean theorem and the rule of three parties. Many examples focus on involving several formulas in one task.