The area of a rhombus is equal to the product. 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 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 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

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 degree measure, 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.

Square 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 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 the rhombus belongs to a separate class of quadrilaterals and stands out among them 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

Perimeter by definition equal to the sum 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 Square root from a quarter of the product of the area and the sine of an 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.

What is Rhombus? A rhombus is a parallelogram in which all sides are equal.

RHOMBUS, a figure on a plane, a quadrilateral with equal sides. Diamond - special case A PARALLELOGRAM in which either two adjacent sides are equal, or the diagonals intersect at right angles, or the diagonal bisects the angle. A rhombus with right angles is called a square.

The classic formula for the area of a rhombus is to calculate the value through the height. The area of a rhombus is equal to the product of a side and the height drawn to that side.

1. The area of a rhombus is equal to the product of a side and the height drawn to this side:

\[ S = a \cdot h \]

2. If the side of a rhombus is known (all sides of a rhombus are equal) and the angle between the sides, then the area can be found using the following formula:

\[ S = a^(2) \cdot sin(\alpha) \]

3. The area of a rhombus is also equal to the half product of the diagonals, that is:

\[ S = \dfrac(d_(1) \cdot d_(2) )(2) \]

4. If the radius r of a circle inscribed in a rhombus and the side of the rhombus a are known, then its area is calculated by the formula:

\[ S = 2 \cdot a \cdot R \]

Properties of a rhombus

In the figure above, \(ABCD\) is a rhombus, \(AC = DB = CD = AD\) . Since a rhombus is a parallelogram, it has all the properties of a parallelogram, but there are also properties inherent only to a rhombus.

You can fit a circle into any rhombus. The center of a circle inscribed in a rhombus is the intersection point of its diagonals. Circle radius equal to half the height of the rhombus:

\[ r = \frac( AH )(2) \]

Properties of a rhombus

The diagonals of a rhombus are perpendicular;

The diagonals of a rhombus are the bisectors of its angles.

Signs of a diamond

A parallelogram whose diagonals intersect at right angles is a rhombus;

A parallelogram whose diagonals are the bisectors of its angles is a rhombus.

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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 opposite sides parallel and opposite angles equal. 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.

Interesting Facts
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. The name came from the same time card suit"diamonds". Very wide diamonds various types used in heraldry.