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 plane figures. One of these shapes is a rhombus. Most problems in solving quadrilaterals come down to finding their areas. Let us systematize known formulas and various methods for 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 triangles. 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 the sine of a 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.
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 us be given a rhombus with diagonals
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: 
A rhombus is a special figure in geometry. Thanks to its special properties, there is not one, but several formulas that can be used to calculate the area of a rhombus. What are these properties and what are the most common formulas for finding the area of this figure? Let's figure it out.
What geometric figure is called a rhombus?
Before you find out what the area of a rhombus is, it’s worth finding out what kind of figure it is.
From the time of Euclidean geometry, a rhombus is a symmetrical quadrilateral, all four sides of which are equal in length and parallel in pairs.
Origin of the term
The name of this figure came to most modern languages from Greek, through the mediation of Latin. The “progenitor” of the word “rhombus” was the Greek noun ῥόμβος (tambourine). Although it is difficult for twentieth-century residents, accustomed to round tambourines, to imagine them in any other shape, among the Hellenes these musical instruments were traditionally made not round, but diamond-shaped.
In most modern languages, this mathematical term is used as in Latin: rombus. However, in English, rhombuses are sometimes called diamond (diamond or diamond). This figure received this nickname because of its special shape, reminiscent of a precious stone. As a rule, a similar term is not used for all rhombuses, but only for those in which the angle of intersection of its two sides is equal to sixty or forty-five degrees.
This figure was first mentioned in the works of the Greek mathematician who lived in the first century of the new era - Heron of Alexandria.
What properties does this geometric figure have?
To find the area of a rhombus, first of all you need to know what features this geometric figure has.
Under what conditions is a parallelogram a rhombus?
As you know, every rhombus is a parallelogram, but not every parallelogram is a rhombus. To accurately state that the presented figure is indeed a rhombus, and not a simple parallelogram, it must correspond to one of the three main features that distinguish a rhombus. Or all three at once.
- The diagonals of a parallelogram intersect at an angle of ninety degrees.
- Diagonals divide angles in two, acting as their bisectors.
- Not only parallel, but also adjacent sides have the same length. This, by the way, is one of the main differences between a rhombus and a parallelogram, since the second figure has only parallel sides that are equal in length, but not adjacent ones.
Under what conditions is a rhombus a square?
According to its properties, in some cases a rhombus can simultaneously become a square. To clearly confirm this statement, simply rotate the square in any direction by forty-five degrees. The resulting figure will be a rhombus, each of the angles of which is equal to ninety degrees.
Also, to confirm that the square is a rhombus, you can compare the characteristics of these figures: in both cases, all sides are equal, and the diagonals are bisectors and intersect at an angle of ninety degrees.
How to find out the area of a rhombus using its diagonals
In the modern world, you can find almost all the materials to perform the necessary calculations on the Internet. Thus, there are a lot of resources equipped with programs for automatically calculating the area of a particular figure. Moreover, if (as in the case of a rhombus) there are several formulas for this, then it is possible to choose which one is most convenient to use. However, first of all, you need to be able to calculate the area of a rhombus yourself without the help of a computer and navigate the formulas. There are many of them for the rhombus, but the most famous of them are four.
One of the simplest and most common ways to find out the area of this figure is if you have information about the length of its diagonals. If the problem has this data, then you can apply the following formula to find the area: S = KM x LN/2 (KM and LN are the diagonals of the rhombus KLMN).
You can check the reliability of this formula in practice. Let's say that a rhombus KLMN has the length of one of its diagonals KM - 10 cm, and the second LN - 8 cm. Then we substitute these data into the above formula and get the following result: S = 10 x 8/ 2 = 40 cm 2.
Formula for calculating the area of a parallelogram
There is another formula. As stated above in the definition of a rhombus, it is not just a quadrilateral, but also a parallelogram, and has all the features of this figure. In this case, to find its area, it is quite advisable to use the formula used for a parallelogram: S = KL x Z. In this case, KL is the length of the side of the parallelogram (rhombus), and Z is the length of the height drawn to this side.
In some problems, the length of the side is not provided, but the perimeter of the rhombus is known. Since the formula for finding it was indicated above, you can use it to find out the length of the side. So, the perimeter of the figure is 10 cm. The length of the side can be found by inverting the perimeter formula and dividing 10 by 4. The result will be 2.5 cm - this is the desired length of the side of the rhombus.
Now it’s worth trying to substitute this number into the formula, knowing that the length of the height drawn to the side is also equal to 2.5 cm. Now let’s try to put these values into the above formula for the area of a parallelogram. It turns out that the area of the rhombus is S = 2.5 x 2.5 = 6.25 cm 2.
Other ways to calculate the area of a rhombus
Those who have already mastered sines and cosines can use formulas containing them to find the area of a rhombus. A classic example is the following formula: S = KM 2 x Sin KLM. In this case, the area of the figure is equal to the product of the two sides of the rhombus multiplied by the sine of the angle between them. And since all the sides in a rhombus are the same, it is easier to immediately square one side, as was shown in the formula.
We check this scheme in practice, and not just for a rhombus, but for a square, which, as you know, has all right angles, which means they are equal to ninety degrees. Let's say one of the sides is 15 cm. It is also known that the sine of an angle of 90° is equal to one. Then, according to the formula, S = 15 x 15 x Sin 90° = 255x1 = 255 cm 2.
In addition to the above, in some cases another formula is used, using sine to determine the area of a rhombus: S = 4 x R 2 /Sin KLM. In this embodiment, the radius of a circle inscribed in a rhombus is used. It is raised to the power of the square and multiplied by four. And the entire result is divided by the sine of the angle closest to the inscribed figure.
As an example, for simplicity of calculations, let’s take a square again (the sine of its angle will always be equal to one). The radius of the circle inscribed in it is 4.4 cm. Then the area of the rhombus will be calculated as follows: S = 4 x 4.4 2 / Sin 90 ° = 77.44 cm 2
The above formulas for finding the radius of a rhombus are far from the only ones of their kind, but they are the easiest to understand and carry out calculations.
A rhombus is a special case of a 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. Diamonds of various types are used very widely in heraldry.
What is Rhombus? A rhombus is a parallelogram with all sides equal.
RHOMBUS, a figure on a plane, a quadrilateral with equal sides. A rhombus is a special case of a PARALLELOGRAM, in which either two adjacent sides are equal, or the diagonals intersect at right angles, or the diagonal bisects the angle. A rhombus with right angles is called a square.
The classic formula for the area of a rhombus is to calculate the value through the height. The area of a rhombus is equal to the product of a side and the height drawn to that side.
1. The area of a rhombus is equal to the product of a side and the height drawn to this side:
\[ S = a \cdot h \]
2. If the side of a rhombus is known (all sides of a rhombus are equal) and the angle between the sides, then the area can be found using the following formula:
\[ S = a^(2) \cdot sin(\alpha) \]
3. The area of a rhombus is also equal to the half product of the diagonals, that is:
\[ S = \dfrac(d_(1) \cdot d_(2) )(2) \]
4. If the radius r of a circle inscribed in a rhombus and the side of the rhombus a are known, then its area is calculated by the formula:
\[ S = 2 \cdot a \cdot R \]
Properties of a rhombus
In the figure above, \(ABCD\) is a rhombus, \(AC = DB = CD = AD\) . Since a rhombus is a parallelogram, it has all the properties of a parallelogram, but there are also properties inherent only to a rhombus.
You can fit a circle into any rhombus. The center of a circle inscribed in a rhombus is the intersection point of its diagonals. Circle radius equal to half the height of the rhombus:
\[ r = \frac( AH )(2) \]
Properties of a rhombus
The diagonals of a rhombus are perpendicular;
The diagonals of a rhombus are the bisectors of its angles.
Signs of a diamond
A parallelogram whose diagonals intersect at right angles is a rhombus;
A parallelogram whose diagonals are the bisectors of its angles is a rhombus.
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