In the school course in geometry, among the main tasks, considerable attention is paid to examples calculating the area and perimeter of a rhombus. Let us remember that a rhombus belongs to a separate class of quadrilaterals and stands out among them by equal sides. A rhombus is also a special case of a parallelogram if the latter has all sides equal AB=BC=CD=AD. Below is a picture showing a rhombus.
Properties of a rhombus
Since a rhombus occupies some part of parallelograms, the properties in them will be similar.
- Opposite angles of a rhombus, like a parallelogram, are equal.
- The sum of the angles of a rhombus adjacent to one side is 180°.
- The diagonals of a rhombus intersect at an angle of 90 degrees.
- The diagonals of a rhombus are also the bisectors of its angles.
- The diagonals of a rhombus are divided in half at the point of intersection.
Signs of a diamond
All the characteristics of a rhombus follow from its properties and help to distinguish it among quadrangles, rectangles, and parallelograms.
- A parallelogram whose diagonals intersect at right angles is a rhombus.
- A parallelogram whose diagonals are bisectors is a rhombus.
- A parallelogram with equal sides is a rhombus.
- A quadrilateral with all sides equal is a rhombus.
- A quadrilateral whose diagonals are angle bisectors and intersect at right angles is a rhombus.
- A parallelogram with equal heights is a rhombus.
Formula for the perimeter of a rhombus
The perimeter by definition is equal to the sum of all sides. Since all sides of a rhombus are equal, we calculate its perimeter using the formula
The perimeter is calculated in units of length.
Radius of a circle inscribed in a rhombus
One of the common problems when studying a rhombus is finding the radius or diameter of the inscribed circle. The figure below shows some of the most common formulas for the radius of an inscribed circle in a rhombus.
The first formula shows that the radius of a circle inscribed in a rhombus is equal to the product of the diagonals divided by the sum of all sides (4a).
Another formula shows that the radius of a circle inscribed in a rhombus is equal to half the height of the rhombus
The second formula in the figure is a modification of the first and is used when calculating the radius of a circle inscribed in a rhombus when the diagonals of the rhombus are known, that is, the unknown sides.
The third formula for the radius of an inscribed circle actually finds half the height of the small triangle that is formed by the intersection of the diagonals.
Among the less popular formulas for calculating the radius of a circle inscribed in a rhombus, you can also give the following:
here D is the diagonal of the rhombus, alpha is the angle that cuts the diagonal.
If the area (S) of a rhombus and the magnitude of the acute angle (alpha) are known, then to calculate the radius of the inscribed circle you need to find the square root of the quarter of the product of the area and the sine of the acute angle: 
From the above formulas you can easily find the radius of a circle inscribed in a rhombus if the conditions of the example contain the required set of data.
Formula for the area of a rhombus
Formulas for calculating area are shown in the figure.
The simplest is derived as the sum of the areas of two triangles into which a rhombus is divided by its diagonal.
The second area formula applies to problems in which the diagonals of a rhombus are known. Then the area of a rhombus is equal to half the product of the diagonals
It is simple enough to remember and also easy to calculate.
The third area formula makes sense when the angle between the sides is known. According to it, the area of a rhombus is equal to the product of the square of the side and the sine of the angle. Whether it is acute or not does not matter since the sine of both angles takes on the same value.
Mathematics is a school subject that is studied by everyone, regardless of class profile. However, she is not everyone's favorite. Sometimes undeservedly. This science constantly presents students with challenges that allow their brains to develop. Mathematics does a great job of keeping children's thinking skills alive. One of its sections copes especially well with this - geometry.
Any of the topics that are studied in it is worthy of attention and respect. Geometry is a way to develop spatial imagination. An example is the topic about the areas of shapes, in particular rhombuses. These puzzles can lead to dead ends if you don't understand the details. Because different approaches to finding the answer are possible. It is easier for some to remember different versions of the formulas that are written below, while others are able to obtain them themselves from previously learned material. In any case, there are no hopeless situations. If you think a little, you will definitely find a solution.
It is necessary to answer this question in order to understand the principles of obtaining formulas and the flow of reasoning in problems. After all, in order to understand how to find the area of a rhombus, you need to clearly understand what kind of figure it is and what its properties are.
For the convenience of considering a parallelogram, which is a quadrilateral with pairwise parallel sides, we will take it as a “parent”. He has two “children”: a rectangle and a rhombus. Both of them are parallelograms. If we continue the parallels, then this is a “surname”. This means that in order to find the area of a rhombus, you can use the already studied formula for a parallelogram.
But, like all children, the rhombus also has something of its own. This makes it slightly different from the "parent" and allows it to be viewed as a separate figure. After all, a rectangle is not a rhombus. Returning to the parallels - they are like brother and sister. They have a lot in common, but they are still different. These differences are their special properties that need to be used. It would be strange to know about them and not apply them in solving problems.
If we continue the analogy and recall another figure - a square, then it will be a continuation of a rhombus and a rectangle. This figure combines all the properties of both.
Properties of a rhombus
There are five of them and they are listed below. Moreover, some of them repeat the properties of a parallelogram, while some are inherent only to the figure in question.
- A rhombus is a parallelogram that has taken on a special shape. It follows from this that its sides are pairwise parallel and equal. Moreover, they are not equal in pairs, but that’s all. As it would be for a square.
- The diagonals of this quadrilateral intersect at an angle of 90º. This is convenient and greatly simplifies the flow of reasoning when solving problems.
- Another property of diagonals: each of them is divided by the point of intersection into equal segments.
- The angles of this figure lying opposite each other are equal.
- And the last property: the diagonals of a rhombus coincide with the bisectors of the angles.
Notations adopted in the considered formulas
In mathematics, you solve problems using common letter expressions called formulas. The topic about squares is no exception.
In order to move on to the notes that will tell you how to find the area of a rhombus, you need to agree on the letters that replace all the numerical values of the elements of the figure.
Now it's time to write the formulas.
The problem data includes only the diagonals of the rhombus
The rule states that to find an unknown quantity, you need to multiply the lengths of the diagonals, and then divide the product in half. The result of division is the area of the rhombus through the diagonals.
The formula for this case will look like this:
Let this formula be number 1.
The problem gives the side of a rhombus and its height
To calculate the area, you will need to find the product of these two quantities. This is perhaps the simplest formula. Moreover, it is also known from the topic about the area of a parallelogram. Such a formula has already been studied there.
Mathematical notation:
The number of this formula is 2.
Known side and acute angle
In this case, you need to square the size of the side of the rhombus. Then find the sine of the angle. And with the third action, calculate the product of the two resulting quantities. The answer will be the area of the rhombus.
Literal expression:
Its serial number is 3.
Given quantities: radius of inscribed circle and acute angle
To calculate the area of a rhombus, you need to find the square of the radius and multiply it by 4. Determine the value of the sine of the angle. Then divide the product by the second quantity.
The formula takes the following form:
It will be numbered 4.
The problem involves the side and radius of an inscribed circle
To determine how to find the area of a rhombus, you will need to calculate the product of these quantities and the number 2.
The formula for this problem will look like this:
Its serial number is 5.
Examples of possible tasks
Problem 1
One of the diagonals of a rhombus is 8 cm, and the other is 14 cm. You need to find the area of the figure and the length of its side.
Solution
To find the first quantity, you will need formula 1, in which D 1 = 8, D 2 = 14. Then the area is calculated as follows: (8 * 14) / 2 = 56 (cm 2).
The diagonals divide the rhombus into 4 triangles. Each of them will definitely be rectangular. This must be used to determine the value of the second unknown. The side of the rhombus will become the hypotenuse of the triangle, and the legs will be the halves of the diagonals.
Then a 2 = (D 1 /2) 2 + (D 2 /2) 2. After substituting all the values, we get: a 2 = (8 / 2) 2 + (14 / 2) 2 = 16 + 49 = 65. But this is the square of the side. This means we need to take the square root of 65. Then the side length will be approximately 8.06 cm.
Answer: area is 56 cm2 and side is 8.06 cm.
Problem 2
The side of a rhombus has a value equal to 5.5 dm, and its height is 3.5 dm. Find the area of the figure.
Solution
In order to find the answer, you will need formula 2. In it, a = 5.5, H = 3.5. Then, replacing the letters in the formula with numbers, we find that the desired value is 5.5 * 3.5 = 19.25 (dm 2).
Answer: The area of a rhombus is 19.25 dm2.
Problem 3
The acute angle of a certain rhombus is 60º, and its smaller diagonal is 12 cm. You need to calculate its area.
Solution
To get the result, you will need formula number 3. In it, instead of A will be 60, and the value A unknown.
To find the side of a rhombus, you will need to remember the theorem of sines. In a right triangle A will be the hypotenuse, the shorter leg is equal to half the diagonal, and the angle is divided in half (known from the property where the bisector is mentioned).
Then the side A will be equal to the product of the leg and the sine of the angle.
The leg needs to be calculated as D/2 = 12/2 = 6 (cm). Sine (A/2) will be equal to its value for an angle of 30º, that is, 1/2.
After performing simple calculations, we obtain the following value for the side of the rhombus: a = 3 (cm).
Now the area is the product of 3 2 and the sine of 60º, that is, 9 * (√3)/2 = (9√3)/2 (cm 2).
Answer: the required value is (9√3)/2 cm 2.
Results: everything is possible
Here we looked at some options for how to find the area of a rhombus. If it is not directly clear in a problem which formula to use, then you need to think a little and try to connect previously studied topics. In other topics there will definitely be a hint that will help connect known quantities with those in the formulas. And the problem will be solved. The main thing is to remember that everything previously learned can and should be used.
In addition to the proposed tasks, inverse problems are also possible, when using the area of a figure you need to calculate the value of some element of a rhombus. Then you need to use the equation that is closest to the condition. And then transform the formula, leaving an unknown quantity on the left side of the equality.
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.
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.
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 a little of the mysterious secrets of the 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 (as opposed to 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 by such famous personalities as 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.
- 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.