Parallelogram in problems. How to find the area of ​​a parallelogram, triangle, trapezoid How to calculate the area of ​​a parallelogram

Geometric area- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface bounded by a 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. Triangle area formula for 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. The formula for the area of ​​a triangle given three sides and the radius of the circumscribed circle
   
3. The formula for the area of ​​a triangle given three sides and the radius of an inscribed circle
   Area of ​​a triangle is equal to the product of the half-perimeter of the triangle and the radius of the inscribed circle.
   
where S is the area of ​​the triangle,
- the lengths of the sides of the triangle,
- the 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. The formula for the area of ​​a square given the length of a side
   square area is equal to the square of its side length.
2. The formula for the area of ​​a square given the length of the diagonal
   square area equal to half the square of the length of its diagonal.
   

   S=	1	2
   	2

where S is the area of ​​the square,
is the length of the side of the square,
is the length of the diagonal of the square.

Rectangle area formula

Rectangle area is equal to the product of the lengths of its two adjacent sides

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

Formulas for the area of ​​a parallelogram

1. Parallelogram area formula for side length and height
   Parallelogram area
2. The formula for the area of ​​a parallelogram given two sides and the angle between them
   Parallelogram area is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.
   

   a b sinα

where S is the area of ​​the parallelogram,
are the lengths of the sides of the parallelogram,
is the height of the parallelogram,
is the angle between the sides of the parallelogram.

Formulas for the area of ​​a rhombus

1. Rhombus area formula given side length and height
   Rhombus area is equal to the product of the length of its side and the length of the height lowered to this side.
2. The formula for the area of ​​a rhombus given the length of the side and the angle
   Rhombus area 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. The formula for the area of ​​a rhombus from the lengths of its diagonals
   Rhombus area is equal to half the product of the lengths of its diagonals.
where S is the area of ​​the rhombus,
- length of the side of the rhombus,
- the length of the height of the rhombus,
- the angle between the sides of the rhombus,
1, 2 - the lengths of the diagonals.

Trapezium area formulas

1. Heron's formula for a trapezoid

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

Parallelogram is a quadrilateral whose sides are pairwise parallel.

In this figure, opposite sides and angles are equal to each other. The diagonals of a parallelogram intersect at one point and bisect it. Parallelogram area formulas allow you to find the value through the sides, height and diagonals. The parallelogram can also be represented in special cases. They are considered a rectangle, square and rhombus.
First, let's consider an example of calculating the area of ​​a parallelogram by height and the side to which it is lowered.

This case is considered a classic and does not require further investigation. It is better to consider the formula for calculating the area through two sides and the angle between them. The same method is used in the calculation. If the sides and the angle between them are given, then the area is calculated as follows:

Suppose we are given a parallelogram with sides a = 4 cm, b = 6 cm. The angle between them is α = 30°. Let's find the area:

Area of ​​a parallelogram in terms of diagonals

The formula for the area of ​​a parallelogram in terms of diagonals allows you to quickly find the value.
For calculations, you need the value of the angle located between the diagonals.

Consider an example of calculating the area of ​​a parallelogram through diagonals. Let a parallelogram be given with diagonals D = 7 cm, d = 5 cm. The angle between them is α = 30°. Substitute the data in the formula:

An example of calculating the area of ​​a parallelogram through a diagonal gave us an excellent result - 8.75.

Knowing the formula for the area of ​​a parallelogram in terms of a diagonal, you can solve many interesting problems. Let's look at one of them.

Task: Given a parallelogram with an area of ​​92 sq. see Point F is located in the middle of its side BC. Let's find the area of ​​the trapezoid ADFB, which will lie in our parallelogram. To begin with, let's draw everything that we received according to the conditions.
Let's get to the solution:

According to our conditions, ah \u003d 92, and accordingly, the area of ​​\u200b\u200bour trapezoid will be equal to

Before we learn how to find the area of ​​a parallelogram, we need to remember what a parallelogram is and what is called its height. A parallelogram is a quadrilateral whose opposite sides are pairwise parallel (lie on parallel lines). The perpendicular drawn from an arbitrary point on the opposite side to the line containing this side is called the height of the parallelogram.

Square, rectangle and rhombus are special cases of parallelogram.

The area of ​​a parallelogram is denoted as (S).

Formulas for finding the area of ​​a parallelogram

S=a*h, where a is the base, h is the height that is drawn to the base.

S=a*b*sinα, where a and b are the bases, and α is the angle between the bases a and b.

S \u003d p * r, where p is the semi-perimeter, r is the radius of the circle that is inscribed in the parallelogram.

The area of ​​the parallelogram formed by the vectors a and b is equal to the modulus of the product of the given vectors, namely:

Consider example No. 1: A parallelogram is given, the side of which is 7 cm, and the height is 3 cm. How to find the area of ​​\u200b\u200bthe parallelogram, we need a formula for solving.

So S= 7x3. S=21. Answer: 21 cm 2.

Consider example No. 2: The bases are 6 and 7 cm, and the angle between the bases is 60 degrees. How to find the area of ​​a parallelogram? Formula used to solve:

Thus, first we find the sine of the angle. Sine 60 \u003d 0.5, respectively S \u003d 6 * 7 * 0.5 \u003d 21 Answer: 21 cm 2.

I hope that these examples will help you in solving problems. And remember, the main thing is knowledge of formulas and attentiveness

What is a parallelogram? A parallelogram is a quadrilateral whose opposite sides are pairwise parallel.

1. The area of ​​a parallelogram is calculated by the formula:

\[ \LARGE S = a \cdot h_(a)\]

Where:
a is the side of the parallelogram,
h a is the height drawn to this side.

2. If the lengths of two adjacent sides of the parallelogram and the angle between them are known, then the area of ​​the parallelogram is calculated by the formula:

\[ \LARGE S = a \cdot b \cdot sin(\alpha) \]

3. If the diagonals of the parallelogram are given and the angle between them is known, then the area of ​​the parallelogram is calculated by the formula:

\[ \LARGE S = \frac(1)(2) \cdot d_(1) \cdot d_(2) \cdot sin(\alpha) \]

Parallelogram properties

In a parallelogram, opposite sides are equal: \(AB = CD \) , \(BC = AD \)

In a parallelogram, opposite angles are: \(\angle A = \angle C \) , \(\angle B = \angle D \)

The diagonals of the parallelogram at the point of intersection are bisected \(AO = OC \) , \(BO = OD \)

The diagonal of a parallelogram divides it into two equal triangles.

The sum of the angles of a parallelogram adjacent to one side is 180 o:

\(\angle A + \angle B = 180^(o) \), \(\angle B + \angle C = 180^(o)\)

\(\angle C + \angle D = 180^(o) \), \(\angle D + \angle A = 180^(o)\)

The diagonals and sides of a parallelogram are related by the following relationship:

\(d_(1)^(2) + d_(2)^2 = 2a^(2) + 2b^(2) \)

In a parallelogram, the angle between the heights is equal to its acute angle: \(\angle K B H =\angle A \) .

Bisectors of angles adjacent to one side of a parallelogram are mutually perpendicular.

Bisectors of two opposite angles of a parallelogram are parallel.

Parallelogram features

A quadrilateral is a parallelogram if:

\(AB = CD \) and \(AB || CD \)

\(AB = CD \) and \(BC = AD \)

\(AO = OC \) and \(BO = OD \)

\(\angle A = \angle C \) and \(\angle B = \angle D \)

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As in Euclidean geometry, the point and the straight line are the main elements of the theory of planes, so the parallelogram is one of the key figures of convex quadrilaterals. From it, like threads from a ball, flow the concepts of "rectangle", "square", "rhombus" and other geometric quantities.

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Definition of a parallelogram

convex quadrilateral, consisting of segments, each pair of which is parallel, is known in geometry as a parallelogram.

What a classic parallelogram looks like is a quadrilateral ABCD. The sides are called the bases (AB, BC, CD and AD), the perpendicular drawn from any vertex to the opposite side of this vertex is called the height (BE and BF), the lines AC and BD are the diagonals.

Attention! Square, rhombus and rectangle are special cases of parallelogram.

Sides and angles: ratio features

Key properties, by and large, predetermined by the designation itself, they are proved by the theorem. These characteristics are as follows:

1. Sides that are opposite are identical in pairs.
2. Angles that are opposite to each other are equal in pairs.

Proof: consider ∆ABC and ∆ADC, which are obtained by dividing quadrilateral ABCD by line AC. ∠BCA=∠CAD and ∠BAC=∠ACD, since AC is common to them (vertical angles for BC||AD and AB||CD, respectively). It follows from this: ∆ABC = ∆ADC (the second criterion for the equality of triangles).

Segments AB and BC in ∆ABC correspond in pairs to lines CD and AD in ∆ADC, which means that they are identical: AB = CD, BC = AD. Thus, ∠B corresponds to ∠D and they are equal. Since ∠A=∠BAC+∠CAD, ∠C=∠BCA+∠ACD, which are also identical in pairs, then ∠A = ∠C. The property has been proven.

Characteristics of the figure's diagonals

Main feature these parallelogram lines: the point of intersection bisects them.

Proof: let m. E be the intersection point of the diagonals AC and BD of the figure ABCD. They form two commensurate triangles - ∆ABE and ∆CDE.

AB=CD since they are opposite. According to lines and secants, ∠ABE = ∠CDE and ∠BAE = ∠DCE.

According to the second sign of equality, ∆ABE = ∆CDE. This means that the elements ∆ABE and ∆CDE are: AE = CE, BE = DE and, moreover, they are commensurate parts of AC and BD. The property has been proven.

Features of adjacent corners

At adjacent sides, the sum of the angles is 180°, since they lie on the same side of the parallel lines and the secant. For quadrilateral ABCD:

∠A+∠B=∠C+∠D=∠A+∠D=∠B+∠C=180º

Bisector properties:

1. , dropped to one side, are perpendicular;
2. opposite vertices have parallel bisectors;
3. the triangle obtained by drawing the bisector will be isosceles.

Determining the characteristic features of a parallelogram by the theorem

The features of this figure follow from its main theorem, which reads as follows: quadrilateral is considered a parallelogram in the event that its diagonals intersect, and this point divides them into equal segments.

Proof: Let lines AC and BD of quadrilateral ABCD intersect in t. E. Since ∠AED = ∠BEC, and AE+CE=AC BE+DE=BD, then ∆AED = ∆BEC (by the first sign of equality of triangles). That is, ∠EAD = ∠ECB. They are also the interior crossing angles of the secant AC for lines AD and BC. Thus, by definition of parallelism - AD || BC. A similar property of the lines BC and CD is also derived. The theorem has been proven.

Calculating the area of ​​a figure

The area of ​​this figure found in several ways one of the simplest: multiplying the height and the base to which it is drawn.

Proof: Draw perpendiculars BE and CF from vertices B and C. ∆ABE and ∆DCF are equal since AB = CD and BE = CF. ABCD is equal to the rectangle EBCF, since they also consist of proportionate figures: S ABE and S EBCD, as well as S DCF and S EBCD. It follows that the area of ​​this geometric figure is the same as that of a rectangle:

S ABCD = S EBCF = BE×BC=BE×AD.

To determine the general formula for the area of ​​a parallelogram, we denote the height as hb, and the side b. Respectively:

Other ways to find area

Area calculations through the sides of the parallelogram and the angle, which they form, is the second known method.

,

Spr-ma - area;

a and b are its sides

α - angle between segments a and b.

This method is practically based on the first, but in case it is unknown. always cuts off a right triangle whose parameters are found by trigonometric identities, i.e. . Transforming the ratio, we get . In the equation of the first method, we replace the height with this product and obtain a proof of the validity of this formula.

Through the diagonals of a parallelogram and an angle, which they create when they intersect, you can also find the area.

Proof: AC and BD intersecting form four triangles: ABE, BEC, CDE and AED. Their sum is equal to the area of ​​this quadrilateral.

The area of ​​each of these ∆ can be found from the expression , where a=BE, b=AE, ∠γ =∠AEB. Since , then a single value of the sine is used in the calculations. That is . Since AE+CE=AC= d 1 and BE+DE=BD= d 2 , the area formula reduces to:

.

Application in vector algebra

The features of the constituent parts of this quadrangle have found application in vector algebra, namely: the addition of two vectors. The parallelogram rule states that if given vectorsAndNotare collinear, then their sum will be equal to the diagonal of this figure, the bases of which correspond to these vectors.

Proof: from an arbitrarily chosen beginning - that is. - we build vectors and . Next, we build a parallelogram OASV, where the segments OA and OB are sides. Thus, the OS lies on the vector or sum.

Formulas for calculating the parameters of a parallelogram

The identities are given under the following conditions:

1. a and b, α - sides and the angle between them;
2. d 1 and d 2 , γ - diagonals and at the point of their intersection;
3. h a and h b - heights lowered to sides a and b;

Parameter	Formula
Finding sides
along the diagonals and the cosine of the angle between them
diagonally and sideways
through height and opposite vertex
Finding the length of the diagonals
on the sides and the size of the top between them
along the sides and one of the diagonals	 Conclusion The parallelogram, as one of the key figures of geometry, is used in life, for example, in construction when calculating the area of ​​\u200b\u200bthe site or other measurements. Therefore, knowledge about the distinguishing features and methods for calculating its various parameters can be useful at any time in life.