Parallelogram is a quadrilateral whose sides are parallel in pairs.
In this figure, opposite sides and angles are equal to each other. The diagonals of a parallelogram intersect at one point and bisect it. Formulas for the area of a parallelogram allow you to find the value using the sides, height and diagonals. A parallelogram can also be presented in special cases. They are considered a rectangle, square and rhombus.
First, let's look at 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 additional investigation. It’s better to consider the formula for calculating the area through two sides and the angle between them. The same method is used in calculations. 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 through diagonals
The formula for the area of a parallelogram using the diagonals allows you to quickly find the value.
For calculations, you will need the size of the angle located between the diagonals.
Let's consider an example of calculating the area of a parallelogram using diagonals. Let a parallelogram be given with diagonals D = 7 cm, d = 5 cm. The angle between them is α = 30°. Let's substitute the data into the formula: 
An example of calculating the area of a parallelogram through the diagonal gave us an excellent result - 8.75.
Knowing the formula for the area of a parallelogram through the diagonal, you can solve many interesting problems. Let's look at one of them.
Task: Given a parallelogram with an area of 92 square meters. 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. First, let's draw everything we received according to the conditions.
Let's get to the solution: 
According to our conditions, ah =92, and accordingly, the area of our 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). A perpendicular drawn from an arbitrary point on the opposite side to a line containing this side is called the height of a 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 =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, which is formed by vectors a and b, is equal to the modulus of the product of the given vectors, namely:
Let's consider example No. 1: Given a parallelogram, the side of which is 7 cm and the height is 3 cm. How to find the area of a parallelogram, we need a formula for the solution.
Thus S= 7x3. S=21. Answer: 21 cm 2.
Consider example No. 2: Given bases are 6 and 7 cm, and also given an angle between the bases of 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 = 0.5, respectively S = 6*7*0.5=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
A parallelogram is a geometric figure that is often found in problems in a geometry course (section planimetry). The key features of this quadrilateral are the equality of opposite angles and the presence of two pairs of parallel opposite sides. Special cases of a parallelogram are rhombus, rectangle, square.
Calculating the area of this type of polygon can be done in several ways. Let's look at each of them.
Find the area of a parallelogram if the side and height are known
To calculate the area of a parallelogram, you can use the values of its side, as well as the length of the height lowered onto it. In this case, the data obtained will be reliable both for the case of a known side - the base of the figure, and if you have at your disposal the side side of the figure. In this case, the required value will be obtained using the formula:
S = a * h (a) = b * h (b),
- S is the area that should have been determined,
- a, b – known (or calculated) side,
- h is the height lowered onto it.
Example: the value of the base of a parallelogram is 7 cm, the length of the perpendicular dropped onto it from the opposite vertex is 3 cm.
Solution:S = a * h (a) = 7 * 3 = 21.
Find the area of a parallelogram if 2 sides and the angle between them are known
Let's consider the case when you know the sizes of two sides of a figure, as well as the degree measure of the angle that they form between themselves. The data provided can also be used to find the area of a parallelogram. In this case, the formula expression will look like this:
S = a * c * sinα = a * c * sinβ,
- a – side,
- c – known (or calculated) base,
- α, β – angles between sides a and c.
Example: the base of a parallelogram is 10 cm, its side is 4 cm less. The obtuse angle of the figure is 135°.
Solution: determine the value of the second side: 10 – 4 = 6 cm.
S = a * c * sinα = 10 * 6 * sin135° = 60 * sin(90° + 45°) = 60 * cos45° = 60 * √2 /2 = 30√2.
Find the area of a parallelogram if the diagonals and the angle between them are known
The presence of known values of the diagonals of a given polygon, as well as the angle that they form as a result of their intersection, allows us to determine the area of the figure.
S = (d1*d2)/2*sinγ,
S = (d1*d2)/2*sinφ,
S is the area to be determined,
d1, d2 – known (or calculated by calculations) diagonals,
γ, φ – angles between diagonals d1 and d2.
Derivation of the formula for the area of a parallelogram comes down to constructing a rectangle equal in area to the given parallelogram. Let us take one side of the parallelogram as the base, and the perpendicular drawn from any point on the opposite side to the straight line containing the base will be called the height of the parallelogram. Then the area of the parallelogram will be equal to the product of its base and its height.
Theorem.The area of a parallelogram is equal to the product of its base and its height.
Proof. Consider a parallelogram with area. Let's take the side as the base and draw the heights (Figure 2.3.1). It is required to prove that.
Figure 2.3.1
Let us first prove that the area of the rectangle is also equal. A trapezoid is made up of a parallelogram and a triangle. On the other hand, it is composed of a rectangle NVSC and a triangle. But right triangles are equal in hypotenuse and acute angle (their hypotenuses are equal as opposite sides of a parallelogram, and angles 1 and 2 are equal as the corresponding angles at the intersection of parallel lines and a transversal), so their areas are equal. Therefore, the areas of the parallelogram and the rectangle are also equal, that is, the area of the rectangle is equal. According to the theorem on the area of a rectangle, but since, then.
The theorem has been proven.
Example 2.3.1.
A circle is inscribed in a rhombus with a side and an acute angle. Determine the area of a quadrilateral whose vertices are the points of contact of the circle with the sides of the rhombus.
Solution:
The radius of a circle inscribed in a rhombus (Figure 2.3.2), since the Quadrangle is a rectangle, since its angles rest on the diameter of the circle. Its area is where (side opposite the angle),. 
Figure 2.3.2
So, 
Answer:
Example 2.3.2.
Given a rhombus whose diagonals are 3 cm and 4 cm. From the vertex of an obtuse angle, the heights are drawn and Calculate the area of the quadrilateral
Solution:
Area of a rhombus (Figure 2.3.3).
So, 
Answer:
Example 2.3.3.
The area of a quadrilateral is Find the area of a parallelogram whose sides are equal and parallel to the diagonals of the quadrilateral.
Solution:
Since and (Figure 2.3.4), then is a parallelogram and, therefore,.
Figure 2.3.4
Similarly, we get from which it follows that.
Answer:.
2.4 Area of a triangle
There are several formulas for calculating the area of a triangle. Let's look at those that are studied at school.
The first formula follows from the formula for the area of a parallelogram and is offered to students in the form of a theorem.
Theorem.The area of a triangle is equal to half the product of its base and height.
Proof. Let be the area of the triangle. Take the side at the base of the triangle and draw the height. Let's prove that:
Figure 2.4.1
Let's build the triangle to a parallelogram as shown in the figure. Triangles are equal on three sides (their common side and the opposite sides of a parallelogram), so their areas are equal. Consequently, the area S of triangle ABC is equal to half the area of the parallelogram, i.e. 
The theorem has been proven.
It is important to draw students' attention to two corollaries that follow from this theorem. Namely:
The area of a right triangle is equal to half the product of its legs.
If the heights of two triangles are equal, then their areas are related as bases.
These two consequences play an important role in solving various kinds of problems. Based on this, another theorem is proved, which has wide application in solving problems.
Theorem. If the angle of one triangle is equal to the angle of another triangle, then their areas are related as the product of the sides enclosing equal angles.
Proof. Let and be the areas of triangles whose angles are equal.
Figure 2.4.2
Let us prove that: 
.
Let's apply a triangle. onto the triangle so that the vertex aligns with the vertex, and the sides overlap the rays respectively.
Figure 2.4.3
Triangles have a common height, so... Triangles also have a common height – therefore,. Multiplying the resulting equalities, we get 
.
The theorem has been proven.
Second formula.The area of a triangle is equal to half the product of its two sides and the sine of the angle between them. There are several ways to prove this formula, and I will use one of them.
Proof. From geometry there is a well-known theorem that the area of a triangle is equal to half the product of the base and the height lowered by this base:
In the case of an acute triangle. In case of an obtuse angle. Ho, and therefore 
. So, in both cases. Substituting in the geometric formula for the area of a triangle, we obtain the trigonometric formula for the area of a triangle:
The theorem has been proven.
Third formula for the area of a triangle - Heron's formula, named after the ancient Greek scientist Heron of Alexandria, who lived in the first century AD. This formula allows you to find the area of a triangle, knowing its sides. It is convenient because it allows you not to make any additional constructions or measure angles. Its conclusion is based on the second of the triangle area formulas we considered and the cosine theorem: and .
Before proceeding with the implementation of this plan, note that
In exactly the same way we have:
Now let’s express the cosine in terms of and:
Since any angle in a triangle is greater and less, then. Means, 
.
Now we separately transform each of the factors in the radical expression. We have:
Substituting this expression into the formula for area, we get:
The topic “Area of a triangle” is of great importance in the school mathematics course. A triangle is the simplest of geometric shapes. It is a “structural element” of school geometry. The vast majority of geometric problems come down to solving triangles. The problem of finding the area of a regular and arbitrary n-gon is no exception.
Example 2.4.1.
What is the area of an isosceles triangle if its base is , and its side is ?
Solution:
-isosceles,
Figure 2.4.4
Let's use the properties of an isosceles triangle - median and height. Then 
According to the Pythagorean theorem:
Finding the area of the triangle:
Answer:
Example 2.4.2.
In a right triangle, the bisector of an acute angle divides the opposite leg into segments 4 and 5 cm long. Determine the area of the triangle.
Solution:
Let (Figure 2.4.5). Then (since BD is a bisector). From here we have 
, that is. Means,
Figure 2.4.5
Answer: 
Example 2.4.3.
Find the area of an isosceles triangle if its base is equal to , and the length of the altitude drawn to the base is equal to the length of the segment connecting the midpoints of the base and the side.
Solution:
According to the condition, – the middle line (Figure 2.4.6). Since we have:
or 
, from hence, 
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. 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
- 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 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
- 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
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
- 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. - 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. 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
- 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,
