Perimeter and area of a parallelogram. How to find the area of a parallelogram

A parallelogram is a quadrangular figure whose opposite sides are parallel and equal in pairs. Its opposite angles are also equal, and the point of intersection of the diagonals of the parallelogram divides them in half, being the center of symmetry of the figure. Special cases of a parallelogram are such geometric shapes as square, rectangle and rhombus. The area of a parallelogram can be found different ways, depending on what initial data accompanies the problem statement.

The key characteristic of a parallelogram, very often used when finding its area, is its height. The height of a parallelogram is usually called a perpendicular drawn from an arbitrary point on the opposite side to a straight segment forming that side.

In the simplest case, the area of a parallelogram is defined as the product of its base and its height.
S = DC ∙ h

where S is the area of the parallelogram;
a - base;
h is the height drawn to the given base.
This formula is very easy to understand and remember if you look at the following figure.
As you can see from this image, if we cut off an imaginary triangle to the left of the parallelogram and attach it to the right, the result will be a rectangle. As you know, the area of a rectangle is found by multiplying its length by its height. Only in the case of a parallelogram will the length be the base, and the height of the rectangle will be the height of the parallelogram lowered to a given side.
The area of a parallelogram can also be found by multiplying the lengths of two adjacent bases and the sine of the angle between them:
S = AD∙AB∙sinα

where AD, AB are adjacent bases forming an intersection point and an angle a between themselves;
α is the angle between the bases AD and AB.
You can also find the area of a parallelogram by dividing in half the product of the lengths of the diagonals of the parallelogram by the sine of the angle between them.
S = ½∙AC∙BD∙sinβ

where AC, BD are the diagonals of the parallelogram;
β is the angle between the diagonals.
There is also a formula for finding the area of a parallelogram through the radius of the circle inscribed in it. It is written as follows:

Area of a parallelogram. In many geometry problems related to the calculation of areas, including tasks on the Unified State Exam, formulas for the area of a parallelogram and a triangle are used. There are several of them, we will look at them here.

It would be too simple to list these formulas; there is already enough of this stuff in reference books and on various websites. I would like to convey the essence - so that you do not cram them, but understand them and can easily remember them at any time. After studying the material in the article, you will understand that there is no need to learn these formulas at all. Objectively speaking, they occur so often in decisions that they remain in memory for a long time.

1. So let's look at a parallelogram. The definition reads:

Why is that? It's simple! To show clearly what the meaning of the formula is, let’s perform some additional constructions, namely, construct the heights:

The area of triangle (2) is equal to the area of triangle (1) - the second sign of equality right triangles"along the leg and hypotenuse." Now let’s mentally “cut off” the second one and move it overlaying it on the first one - we get a rectangle, the area of which will be equal to the area of the original parallelogram:

The area of a rectangle is known to be equal to the product of its adjacent sides. As can be seen from the sketch, one side of the resulting rectangle is equal to the side of the parallelogram, and the other is equal to the height of the parallelogram. Therefore, we obtain the formula for the area of a parallelogram S = a∙h a

2. Let's continue, another formula for its area. We have:

Area of a parallelogram formula

Let's denote the sides as a and b, the angle between them is γ "gamma", the height is h a. Consider a right triangle:

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,

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 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 started with the solution:

According to our conditions, ah =92, and accordingly, the area of our trapezoid will be equal to