The area of ​​the rhombus: formulas and facts

The rhombus (from the ancient Greek ῥόμβος and from Latinrombus "tambourine") is a parallelogram, which is characterized by the presence of identical sides along the length. In the case where the angles are 90 degrees (or a right angle), such a geometric figure is called a square. A rhombus is a geometric figure, a kind of quadrangles. It can be both a square and a parallelogram.

Origin of the term

Let's talk a little about the history of this figure, thatwill help us to uncover for ourselves the mysterious secrets of the ancient world. A word familiar to us, often found in school literature, is a "rhombus", originating from the ancient Greek word "tambourine". In Ancient Greece, these musical instruments were made in the form of a diamond or a square (in contrast to modern devices). Surely you noticed that the card suit - tambourine - has a rhombic shape. The formation of this suit goes back to those times when round tambourines were not used in everyday life. Consequently, the rhombus is the oldest historical figure, which was invented by mankind long before the appearance of the wheel.

For the first time such a word as the "rhombus" was used by such well-known personalities as Geron and Pope of Alexandria.

Properties of diamond

1. Since the sides of the diamond oppose each other and are pairwise parallel, the diamond is undoubtedly a parallelogram (AB || CD, AD || BC).
2. Rhombic diagonals have an intersection at right angles (AC ⊥ BD), and, hence, are perpendicular. Therefore, the intersection divides the diagonals in half.
3. Bisectors of rhombic angles are the rhombic diagonals (∠DCA = ∠BCA, ∠ABD = ∠CBD, etc.).
4. From the identity of parallelograms it follows that the sum of all the squares of the diamonds of the rhombus is the number of the side square, which is multiplied by 4.

Signs of a diamond

The rhombus in those cases is a parallelogram when it meets the following conditions:

1. All sides of the parallelogram are equal.
2. The diagonal of the rhombus intersects the right angle, that is, they are perpendicular to each other (AC⊥BD). This proves the rule of the three sides (the sides are equal and are at an angle of 90 degrees).
3. The diagonals of the parallelogram divide the angles equally, since the sides are equal.

Diamond Square

The rhombus area can be calculated by means of several formulas (depending on the material provided in the task). Further, read about what the area of ​​the diamond is equal to.

1. The area of ​​the diamond is equal to the number, which is half the product of all its diagonals.
2. 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 at its height (h).
3. In addition, the area of ​​the rhombus can be calculated froma formula that is the product of the squared side of the rhombus on the sine of the angle. The sine of the angle is the alpha angle between the sides of the original rhombus.
4. Quite acceptable for the correct solution is the formula, which is the product of the doubled alpha angle and the radius of the inscribed circle (r).

You can calculate and prove these formulas on the basis of the Pythagorean theorem and the rule of three parties. Many examples are focused on using several formulas in one job.