How to find the area of ​​a rhombus by side. Four formulas that can be used to calculate the area of ​​a rhombus

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 areas of which will add up to 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 get 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

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:

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 to reveal a little 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

1. 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).
2. Rhombic diagonals intersect at right angles (AC ⊥ BD), and therefore are perpendicular. Therefore, the intersection bisects the diagonals.
3. The bisectors of rhombic angles are the diagonals of the rhombus (∠DCA = ∠BCA, ∠ABD = ∠CBD, etc.).
4. 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:

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

Area of ​​a rhombus

1. The area of ​​a rhombus is equal to the number that is half the product of all its diagonals.
2. 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).
3. 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.
4. 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.

In the article we will consider rhombus area formula and not just one! We'll show you in the pictures how easy it is to be area of ​​a rhombus using simple formulas.

There are a large number of tasks for finding one or another quantity in a rhombus, and the formulas that will be discussed will help us with this.
A rhombus is a separate type of quadrilateral because all its sides are equal. It also represents a special case of a parallelogram in which the sides AB=BC=CD=AD are equal.

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A rhombus has the following properties:

A rhombus has equal parallel angles
- the addition of two adjacent angles is equal to 180 degrees,
- Intersection of diagonals at an angle of 90 degrees,
- The bisectors of a rhombus are its diagonals,
- When intersecting, the diagonal is divided into equal parts.

A rhombus has the following characteristics:

If a parallelogram in which the diagonals meet at an angle of 90 degrees, then it is called a rhombus.
- If a parallelogram whose bisector is a diagonal, then it is called a rhombus.
- If a parallelogram has equal sides, it is a rhombus.
- If a quadrilateral has equal sides, it is a rhombus.
- If a quadrilateral in which the bisector is a diagonal and the diagonals meet at an angle of 90 degrees, then it is a rhombus.
- If a parallelogram has the same heights, it is a rhombus.

From the above signs we can conclude that they are needed in order to learn to separate a rhombus from other figures similar to it.

Because in a rhombus all sides are the same the perimeter is according to the following formula:
P=4a
Area of ​​rhombus formula

There are several formulas. The simplest one is solved by adding the area of ​​2 triangles, which were obtained by dividing the diagonals.

Using the second formula, you can solve problems with known diagonals of a rhombus. In this case, the area of ​​the rhombus will be: the sum of the diagonals divided by two.

It is very simple to solve and will not be forgotten.

The third formula can be used when you know the angle between the sides. Knowing it, you can find the area of ​​a rhombus; it will be equal to the square of the sides times the sine of the angle. It makes no difference what angle. since the sine of an angle has the same value.

It is important to remember that area is measured in squares, and perimeter is measured in units. These formulas are very easy to apply in practice.

You may also encounter problems involving finding the radius of a circle inscribed in a rhombus.

There are also several formulas for this:

Using the first formula, the radius is found as the product of the diagonals divided by the number obtained from the addition of all sides. or equal to half the height (r=h/2).

The second formula takes the principle from the first and applies we know the diagonals and sides of a rhombus.

In the third formula, the radius comes from the height of the smaller triangle resulting from the intersection.

Definition of a diamond

Rhombus is a parallelogram in which all sides are equal to each other.

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If the sides of a rhombus form a right angle, then we get square.

The diagonals of a rhombus intersect at right angles.
The diagonals of a rhombus are the bisectors of its angles.

The area of ​​a rhombus, like the areas of most geometric shapes, can be found in several ways. Let's understand their essence and consider examples of solutions.

Formula for the area of ​​a rhombus by side and height

Let us be given a rhombus with a side a a a and height h h h, drawn to this side. Since a rhombus is a parallelogram, we find its area in the same way as the area of ​​a parallelogram.

S = a ⋅ h S=a\cdot h S=a ⋅h

A a a- side;
h h h- height lowered to the side a a a.

Let's solve a simple example.

Example

The side of a rhombus is 5 (cm). The height lowered to this side has a length of 2 (cm). Find the area of ​​a rhombus S S S.

Solution

A = 5 a=5 a =5
h = 2 h=2 h =2

We use our formula and calculate:
S = a ⋅ h = 5 ⋅ 2 = 10 S=a\cdot h=5\cdot 2=10S=a ⋅h =5 ⋅ 2 = 1 0 (see sq.)

Answer: 10 cm sq.

Formula for the area of ​​a rhombus using diagonals

Everything is just as simple here. You just need to take half the product of the diagonals and get the area.

S = 1 2 ⋅ d 1 ⋅ d 2 S=\frac(1)(2)\cdot d_1\cdot d_2S=2 1 ​ ⋅ d 1 ​ ⋅ d 2 ​

D 1, d 2 d_1, d_2 d 1 ​ , d 2 ​ - diagonals of a rhombus.

Example

One of the diagonals of a rhombus is 7 (cm), and the other is 2 times larger than the first. Find the area of ​​the figure.

Solution

D 1 = 7 d_1=7 d 1 ​ = 7
d 2 = 2 ⋅ d 1 d_2=2\cdot d_1d 2 ​ = 2 ⋅ d 1 ​

Let's find the second diagonal:
d 2 = 2 ⋅ d 1 = 2 ⋅ 7 = 14 d_2=2\cdot d_1=2\cdot 7=14d 2 ​ = 2 ⋅ d 1 ​ = 2 ⋅ 7 = 1 4
Then the area:
S = 1 2 ⋅ 7 ⋅ 14 = 49 S=\frac(1)(2)\cdot7\cdot14=49S=2 1 ​ ⋅ 7 ⋅ 1 4 = 4 9 (see sq.)

Answer: 49 cm sq.

Formula for the area of ​​a rhombus using two sides and the angle between them

S = a 2 ⋅ sin ⁡ (α) S=a^2\cdot\sin(\alpha)S=a 2 ⋅ sin(α)

A a a- side of the rhombus;
α\alpha α - any angle of the rhombus.

Example

Find the area of ​​a rhombus if each of its sides is 10 cm and the angle between two adjacent sides is 30 degrees.

Solution

A = 10 a=10 a =1 0
α = 3 0 ∘ \alpha=30^(\circ)α = 3 0 ∘

Using the formula we get:
S = a 2 ⋅ sin ⁡ (α) = 100 ⋅ sin ⁡ (3 0 ∘) = 50 S=a^2\cdot\sin(\alpha)=100\cdot\sin(30^(\circ))= 50S=a 2 ⋅ sin(α) =1 0 0 ⋅ sin (3 0 ∘ ) = 5 0 (see sq.)

Answer: 50 cm sq.

Formula for the area of ​​a rhombus based on the radius of the inscribed circle and angle

S = 4 ⋅ r 2 sin ⁡ (α) S=\frac(4\cdot r^2)(\sin(\alpha))S=sin(α)4 ⋅ r 2 ​

R r r- radius of the inscribed circle in a rhombus;
α\alpha α - any angle of the rhombus.

Example

Find the area of ​​a rhombus if the angle between the bases is 60 degrees and the radius of the inscribed circle is 4 (cm).

Solution

R = 4 r=4 r =4
α = 6 0 ∘ \alpha=60^(\circ)α = 6 0 ∘

S = 4 ⋅ r 2 sin ⁡ (α) = 4 ⋅ 16 sin ⁡ (6 0 ∘) ≈ 73.9 S=\frac(4\cdot r^2)(\sin(\alpha))=\frac(4\ cdot 16)(\sin(60^(\circ)))\approx73.9S=sin(α)4 ⋅ r 2 ​ = sin (6 0 ∘ ) 4 ⋅ 1 6 ​ ≈ 7 3 . 9 (see sq.)

Answer: 73.9 cm sq.

Formula for the area of ​​a rhombus based on the radius of the inscribed circle and side

S = 2 ⋅ a ⋅ r S=2\cdot a\cdot rS=2 ⋅ a ⋅r

A a a-side of the rhombus;
r r r- radius of the inscribed circle in a rhombus.

Example

Let's take the condition from the previous problem, but let us instead of the angle know the side of the rhombus equal to 5 cm.

Solution

A = 5 a=5 a =5
r = 4 r=4 r =4

S = 2 ⋅ a ⋅ r = 2 ⋅ 5 ⋅ 4 = 40 S=2\cdot a\cdot r=2\cdot5\cdot4=40S=2 ⋅ a ⋅r =2 ⋅ 5 ⋅ 4 = 4 0 (see sq.)

Answer: 40 cm sq.