Area of a rhombus formula by sides. Four formulas that can be used to calculate the area of a rhombus. Properties of a rhombus
A rhombus (from the ancient Greek ῥόμβος and from the Latin rombus “tambourine”) is a parallelogram, which is characterized by the presence of sides of equal length. When the angles are 90 degrees (or a right angle), such a geometric figure is called a square. Diamond - geometric figure, a type of quadrilateral. It can be both a square and a parallelogram.
Origin of this term
Let's talk a little about the history of this figure, which will help to reveal a little for ourselves mysterious secrets ancient world. A familiar word for us, often found in school literature, "rhombus", originates from ancient Greek word"tambourine". IN Ancient Greece these musical instruments were produced in the shape of a diamond or square (unlike modern devices). Surely you noticed that the card suit - diamonds - has a rhombic shape. The formation of this suit goes back to the times when round diamonds were not used in everyday life. Consequently, the rhombus is the oldest historical figure that was invented by mankind long before the advent of the wheel.
For the first time such a word as “rhombus” was used so famous personalities, like Heron and the Pope of Alexandria.
Properties of a rhombus
- Since the sides of a rhombus are opposite each other and are parallel in pairs, then the rhombus is undoubtedly a parallelogram (AB || CD, AD || BC).
- Rhombic diagonals intersect at right angles (AC ⊥ BD), and therefore are perpendicular. Therefore, the intersection bisects the diagonals.
- The bisectors of rhombic angles are the diagonals of the rhombus (∠DCA = ∠BCA, ∠ABD = ∠CBD, etc.).
- From the identity of parallelograms it follows that the sum of all the squares of the diagonals of a rhombus is the number of the square of the side, which is multiplied by 4.
Signs of a diamond
A rhombus is a parallelogram when it meets the following conditions:
- All sides of a parallelogram are equal.
- The diagonals of a rhombus intersect 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 at an angle of 90 degrees).
- The diagonals of a parallelogram divide the angles equally because the sides are equal.
Area of a rhombus
- The area of a rhombus is equal to the number that is half the product of all its diagonals.
- Since a rhombus is a kind of parallelogram, the area of the rhombus (S) is the product of the side of the parallelogram and its height (h).
- In addition, the area of a rhombus can be calculated using the formula, which is the product of the squared side of the rhombus and the sine of the angle. The sine of the angle is alpha - the angle located between the sides of the original rhombus.
- Quite acceptable for the right decision the formula is considered to be the product of twice the angle alpha and the radius of the inscribed circle (r).
is a parallelogram in which all sides are equal, then all the same formulas apply to it as for a parallelogram, including the formula for finding the area through the product of height and sides.
The area of a rhombus can be found by also knowing its diagonals. The diagonals divide the rhombus into four absolutely identical right triangles. If we sort them to get a rectangle, then its length and width will be equal to one whole diagonal and half of the second diagonal. Therefore, the area of a rhombus is found by multiplying the diagonals of the rhombus, reduced by two (as the area of the resulting rectangle).
If you only have an angle and a side at your disposal, then you can use the diagonal as an assistant and draw it opposite the known angle. Then it will divide the rhombus into two congruent triangles, the total areas of which will give us the area of the rhombus. The area of each of the triangles will be equal to half the product of the square of the side and the sine of the known angle, as the area of an isosceles triangle. Since there are two such triangles, the coefficients are reduced, leaving only the side to the second power and the sine:
If you inscribe a circle inside a rhombus, then its radius will relate to the side at an angle of 90°, which means that twice the radius will be equal to the height of the rhombus. Substituting instead of height h=2r into the previous formula, we obtain area S=ha=2ra
If, along with the radius of the inscribed circle, not a side, but an angle is given, then you must first find the side by drawing the height in such a way as to obtain a right triangle with a given angle. Then side a can be found from trigonometric relations using the formula 
. Substituting this expression into the same standard formula for the area of a rhombus, we get 
Rhombus is special case parallelogram. It is a flat quadrangular figure in which all sides are equal. This property determines that rhombuses have parallel opposite sides and equal opposite angles. The diagonals of a rhombus intersect at right angles, the point of their intersection is in the middle of each diagonal, and the angles from which they emerge are divided in half. That is, they diagonals of a rhombus are bisectors of the angles. Based on the above definitions and the listed properties of rhombuses, their area can be determined in various ways.
1. If both diagonals of a rhombus AC and BD are known, then the area of the rhombus can be determined as half the product of the diagonals.
S = ½ ∙ A.C. ∙ BD
where AC, BD are the length of the diagonals of the rhombus.
To understand why this is so, you can mentally fit a rectangle into a rhombus so that the sides of the latter are perpendicular to the diagonals of the rhombus. It becomes obvious that the area of the rhombus will be equal to half the area of the rectangle inscribed in this way into the rhombus, the length and width of which will correspond to the size of the diagonals of the rhombus.
2. By analogy with a parallelepiped, the area of a rhombus can be found as the product of its side and the height of the perpendicular from the opposite side lowered to a given side.
S = a ∙ h
where a is the side of the rhombus;
h is the height of the perpendicular dropped to a given side.
3. The area of a rhombus is also equal to the square of its side multiplied by the sine of the angle α.
S = a 2 ∙ sin α
where a is the side of the rhombus;
α is the angle between the sides.
4. Also, the area of a rhombus can be found through its side and the radius of the circle inscribed in it.
S=2 ∙ a ∙ r
where a is the side of the rhombus;
r is the radius of the circle inscribed in the rhombus.
The word rhombus comes from the ancient Greek rombus, which means “tambourine”. In those days, tambourines actually had a diamond shape, and not round, as we are used to seeing them now. From the same time, the name of the card suit “diamonds” came about. Very wide diamonds various types used in heraldry.
is a parallelogram in which all sides are equal.
A rhombus with right angles is called a square and is considered a special case of a rhombus. You can find the area of a rhombus in various ways, using all its elements - sides, diagonals, height. The classic formula for the area of a rhombus is to calculate the value through the height.
An example of calculating the area of a rhombus using this formula is very simple. You just need to substitute the data and calculate the area.
Area of a rhombus through diagonals
The diagonals of a rhombus intersect at right angles and are divided in half at the intersection point.
The formula for the area of a rhombus through its diagonals is the product of its diagonals divided by 2.
Let's look at an example of calculating the area of a rhombus using diagonals. Let a rhombus with diagonals be given
d1 =5 cm and d2 =4. Let's find the area. 
The formula for the area of a rhombus through the sides also implies the use of other elements. If a circle is inscribed in a rhombus, then the area of the figure can be calculated from the sides and its radius:
An example of calculating the area of a rhombus through the sides is also very simple. You only need to calculate the radius of the inscribed circle. It can be derived from the Pythagorean theorem and using the formula.
Area of a rhombus through side and angle
The formula for the area of a rhombus in terms of side and angle is used very often.
Let's look at an example of calculating the area of a rhombus using a side and an angle.
Task: Given a rhombus whose diagonals are d1 = 4 cm, d2 = 6 cm. The acute angle is α = 30°. Find the area of the figure using the side and angle.
First, let's find the side of the rhombus. We use the Pythagorean theorem for this. We know that at the point of intersection the diagonals bisect and form a right angle. Hence: 
Let's substitute the values:
Now we know the side and angle. Let's find the area: 
Despite the fact that mathematics is the queen of sciences, and arithmetic is the queen of mathematics, geometry is the most difficult thing for schoolchildren to learn. Planimetry is a branch of geometry that studies flat figures. One of these shapes is a rhombus. Most problems in solving quadrilaterals come down to finding their areas. Let's systematize famous formulas And various ways calculating the area of a rhombus.
A rhombus is a parallelogram with all four sides equal. Recall that a parallelogram has four angles and four pairs of parallel equal sides. Like any quadrilateral, a rhombus has a number of properties, which boil down to the following: when the diagonals intersect, they form an angle equal to 90 degrees (AC ⊥ BD), the intersection point divides each into two equal segments. The diagonals of a rhombus are also the bisectors of its angles (∠DCA = ∠BCA, ∠ABD = ∠CBD, etc.). It follows that they divide the rhombus into four equal right triangle. The sum of the lengths of the diagonals raised to the second power is equal to the length of the side to the second power multiplied by 4, i.e. BD 2 + AC 2 = 4AB 2. There are many methods used in planimetry to calculate the area of a rhombus, the application of which depends on the source data. If the side length and any angle are known, you can use the following formula: the area of a rhombus is equal to the square of the side multiplied by the sine of the angle. From the trigonometry course we know that sin (π – α) = sin α, which means that in calculations you can use the sine of any angle - both acute and obtuse. A special case is a rhombus, in which all angles are right. This is a square. It is known that sine right angle is equal to one, so the area of a square is equal to the length of its side raised to the second power.
As you can see, there are many ways to find the area of a rhombus. Of course, to remember each of them will require patience, attentiveness and, of course, time. But in the future, you can easily choose the method suitable for your task, and you will find that geometry is not difficult.