17.02.2023

Area of ​​a rhombus examples. Area of ​​a rhombus

Area of ​​a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited by the closed contour of this figure). The size of the area is expressed by the number of square units contained in it.

Triangle area formulas

  1. Formula for the area of ​​a triangle by side and height
    Area of ​​a triangle equal to half the product of the length of a side of a triangle and the length of the altitude drawn to this side
  2. Formula for the area of ​​a triangle based on three sides and the radius of the circumcircle
  3. Formula for the area of ​​a triangle based on three sides and the radius of the inscribed circle
    Area of ​​a triangle is equal to the product of the semi-perimeter of the triangle and the radius of the inscribed circle.
    4. where S is the area of ​​the triangle,
    - lengths of the sides of the triangle,
    - height of the triangle,
    - the angle between the sides and,
    - radius of the inscribed circle,
    R - radius of the circumscribed circle,

Square area formulas

  1. Formula for the area of ​​a square by side length
    Square area equal to the square of the length of its side.
  2. Formula for the area of ​​a square along the diagonal length
    Square area equal to half the square of the length of its diagonal.
    S=1 2
    2
    3. where S is the area of ​​the square,
    - length of the side of the square,
    - length of the diagonal of the square.

Rectangle area formula

    Area of ​​a rectangle equal to the product of the lengths of its two adjacent sides

    where S is the area of ​​the rectangle,
    - lengths of the sides of the rectangle.

Parallelogram area formulas

  1. Formula for the area of ​​a parallelogram based on side length and height
    Area of ​​a parallelogram
  2. Formula for the area of ​​a parallelogram based on two sides and the angle between them
    Area of ​​a parallelogram is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.

    a b sin α

    3. where S is the area of ​​the parallelogram,
    - lengths of the sides of the parallelogram,
    - length of parallelogram height,
    - the angle between the sides of the parallelogram.

Formulas for the area of ​​a rhombus

  1. Formula for the area of ​​a rhombus based on side length and height
    Area of ​​a rhombus equal to the product of the length of its side and the length of the height lowered to this side.
  2. Formula for the area of ​​a rhombus based on side length and angle
    Area of ​​a rhombus is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus.
  3. Formula for the area of ​​a rhombus based on the lengths of its diagonals
    Area of ​​a rhombus equal to half the product of the lengths of its diagonals.
    4. where S is the area of ​​the rhombus,
    - length of the side of the rhombus,
    - length of the height of the rhombus,
    - the angle between the sides of the rhombus,
    1, 2 - lengths of diagonals.

Trapezoid area formulas

  1. Heron's formula for trapezoid

    Where S is the area of ​​the trapezoid,
    - lengths of the bases of the trapezoid,
    - lengths of the sides of the trapezoid,

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. Also represents special case parallelogram in which 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 easy 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.

Online calculator

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.

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 in terms of 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:

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.

If the size of the sides is unknown, we use the length of the diagonals. In this case, the area of ​​the rhombus is equal to half the product of the major and minor diagonals.

Given the known length of the diagonals and the size of any angle, the area of ​​a rhombus is determined in two ways. First: area is half the square of the larger diagonal multiplied by the tangent of half the degree measure 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 lowered 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.

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