How to find the area of a rhombus if the side is known. How to find the area of a rhombus

25.04.2019 Education

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 the known 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 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.

Given the known length of the diagonals and the size of any angle, the area of a rhombus is determined in two ways. First: the area is half the square of the larger diagonal, multiplied by the tangent of half the degree measure of the acute angle, i.e. S = 1/2*D 2 *tg(α/2), where D is the major diagonal, α is the acute angle. If you know the size of the minor diagonal, we will use the formula 1/2*d 2 *tg(β/2), where d is the minor diagonal, β is an obtuse angle. Let us recall that the measure of an acute angle is less than 90 degrees (the measure of a right angle), and an obtuse angle, accordingly, is greater than 90 0.

The area of a rhombus can be found using the length of the side (remember, all sides of a rhombus are equal) and height. Height is a perpendicular dropped to the side opposite the angle or to its extension. In order for the base of the height to be located inside the rhombus, it should be lowered from an obtuse angle.

Sometimes a problem requires finding the area of a rhombus based on data related to the inscribed circle. In this case, you need to know its radius. There are two formulas that can be used for calculation. So, to answer the question, you can double the product of the side of the rhombus and the radius of the inscribed circle. In other words, you need to multiply the diameter of the inscribed circle by the side of the rhombus. If the magnitude of the angle is presented in the problem statement, then the area is found through the quotient between the square of the radius multiplied by four and the sine of the angle.

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.

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 language Sometimes rhombuses are called diamond (diamond or diamond). This figure received this nickname because of its special shape, reminiscent of gem. 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 writings of a Greek mathematician who lived in the first century. 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.

As already mentioned in the definition of a rhombus, it is a quadrilateral. And for the reason that its opposite sides are parallel to each other in pairs, a rhombus can also be called a parallelogram, which means that most of the properties of this figure apply to it. Both diagonals of a rhombus at the point of their intersection are evenly divided in two. And due to the fact that they intersect at an angle of ninety degrees, the diagonals divide the figure into 4 right triangles. In any rhombus, the diagonals divide its angles in two, being at the same time their bisectors. If each of the two diagonals of the rhombus is raised to the power of a square, then they the sum will be equal to the product of the square of the side of this figure and the number four. If you connect the midpoints of the four sides of a rhombus with lines, the resulting figure will turn out to be a rectangle. If a circle is inscribed in a rhombus (regardless of its angles), then its center point will coincide with the center of intersection of the diagonals. Diagonals in a rhombus touch the axes of its symmetry at angles of ninety degrees. Since all sides of a rhombus are identical in length, its perimeter is calculated by the formula P = 4 x K (K is the length of one of the sides).

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.

parallel sides

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 modern world On the Internet you can find almost all the materials to perform the necessary calculations. 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 we get the following result: S = 10 x 8/ 2 = 40 cm2.

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 cm2.

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 = KM2 x Sin KLM. IN 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 cm2.

In addition to the above, in some cases another formula is used, using sine to determine the area of a rhombus: S = 4 x R2/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 4.42 / Sin 90 ° = 77.44 cm2

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 (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. A rhombus is a 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 us uncover some mysterious secrets for ourselves. ancient world. The familiar word for us, often found in school literature, “rhombus,” originates from the ancient Greek word “tambourine.” IN Ancient Greece these musical instruments were produced in a diamond or square shape (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 can be calculated using several formulas (depending on the material provided in the problem). Next, read about what the area of a rhombus is.

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.
A formula that is the product of twice the angle alpha and the radius of the inscribed circle (r) is considered quite acceptable for the correct solution.

Instructions

To determine the area, you need to familiarize yourself with a small list of properties belonging to a rhombus:
- opposite angles are always equal;

The diagonals are perpendicular to each other;

Also, the diagonals at the intersection point are divided in half;

Diagonals bisect angles, so they are also bisectors;

Angles adjacent to one side add up to 180°;

It was written in detail about the diagonals of a rhombus, which is not in vain, because they are used in the formula for finding the area.
The first formula: S=d1*d2/2, where d1,d2 are the diagonals of the rhombus.

The second formula uses the angle of the rhombus adjacent to one of the sides, which is also used in the calculation.

S=a*2sin(α), where a is the side of the rhombus; α is the angle between the sides of the rhombus. Finding the sine from a given angle will not be difficult if you have a calculator at hand or you find the values in a special table of sines.

The formula for calculating the area of a rhombus containing the sine of the angle is not the only one. There is the following way:
S=4r^2/sin(α). All values are known and understandable, except for the one that appears - this is the maximum radius of the circle that can fit in the figure.

And the last formula:

S=a*H, where a, as specified in advance, is a side; H is the height of the rhombus.

If all sides are flat geometric figure with parallel opposite sides(parallelogram) are equal, the diagonals intersect at an angle of 90° and bisect the angles at the vertices of the polygon, then it can be called a rhombus. These additional properties of a quadrilateral greatly simplify the formula for finding its area.

Instructions

If the lengths of both diagonals of a rhombus (E and F) are known, then to find the area of the figure (S), calculate the value of half the product of these two quantities: S=½*E*F.

If in the conditions of the problem the length of one of the sides (A) is given, as well as the height (h) of this geometric figure, then to find the area (S) use the formula that applies to all parallelepipeds. The height is a segment perpendicular to the side, connecting it to one of the vertices of the rhombus. The formula for calculating the area using this data is very simple - they must be multiplied: S=A*h.