Area of a rhombus by diagonals and sides. How to find the area of a rhombus

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 flat figures. One of these shapes is a rhombus. Most problems in solving quadrilaterals come down to finding their areas. Let's systematize famous formulas and various ways to calculate 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 triangle. 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 sine right angle equal to one, therefore 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: 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 acute angle. If you know the size of the smaller diagonal, use the formula 1/2*d 2 *tg(β/2), where d is the smaller diagonal, β is 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 the 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.

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

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
Formula for the area of a triangle based on three sides and the radius of the circumcircle
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.

Square area formulas

Formula for the area of a square by side length
Square area equal to the square of the length of its side.
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

Rectangle area formula

Area of a rectangle

Parallelogram area formulas

Formula for the area of a parallelogram based on side length and height
Area of a parallelogram
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

Formulas for the area of a rhombus

Formula for the area of a rhombus based on side length and height
Area of a rhombus is equal to the product of the length of its side and the length of the height lowered to this side.
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.
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.

Trapezoid area formulas

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.
Diamond refers to separate species quadrilaterals, since all 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 same heights- this 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.

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:

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. Diamond - 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. A familiar word for us, often found in school literature, “diamond,” originates from the ancient Greek word “tambourine.” IN Ancient Greece these musical instruments were produced in the shape of a diamond or square (unlike modern devices). Surely you noticed that card suit- tambourine - 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 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.
Quite acceptable for the right decision the formula is considered to be the product of twice the angle alpha and the radius of the inscribed circle (r).