Theorem 1.

The area of a square is equal to the square of its side.

Let us prove that the area S of a square with side a is equal to a 2. Let's take a square with side 1 and divide it into n equal squares as shown in Figure 1. geometry area figure theorem

Picture 1.

Since the side of a square is 1, then the area of each small square is equal. The side of each small square is equal, i.e. equal to a. It follows that. The theorem has been proven.

Theorem 2.

The area of a parallelogram is equal to the product of its side and the height drawn to this side (Fig. 2.):

S = a * h.

Let ABCD be the given parallelogram. If it is not a rectangle, then one of its corners A or B is acute. For definiteness, let angle A be acute (Fig. 2).

Figure 2.

Let us drop a perpendicular AE from vertex A to line CB. The area of the trapezoid AECD is equal to the sum of the areas of the parallelogram ABCD and the triangle AEB. Let us drop a perpendicular DF from vertex D to line CD. Then the area of the trapezoid AECD is equal to the sum of the areas of the rectangle AEFD and the triangle DFC. Right triangles AEB and DFC are congruent and therefore have equal areas. It follows that the area of parallelogram ABCD is equal to the area of rectangle AEFD, i.e. equals AE * AD. Segment AE is the height of the parallelogram lowered to side AD, and therefore S = a * h. The theorem has been proven.

Theorem 3

The area of a triangle is equal to half the product of its side and its altitude(Fig. 3.):

Figure 3.

Proof.

Let ABC be the given triangle. Let's add it to parallelogram ABCD, as shown in the figure (Fig. 3.1.).

Figure 3.1.

The area of a parallelogram is equal to the sum of the areas of triangles ABC and CDA. Since these triangles are congruent, the area of the parallelogram is equal to twice the area of triangle ABC. The height of the parallelogram corresponding to side CB is equal to the height of the triangle drawn to side CB. This implies the statement of the theorem. The theorem is proven.

Theorem 3.1.

The area of a triangle is equal to half the product of its two sides and the sine of the angle between them(Figure 3.2.).

Figure 3.2.

Proof.

Let us introduce a coordinate system with the origin at point C so that B lies on the positive semi-axis C x, and point A has a positive ordinate. The area of a given triangle can be calculated using the formula, where h is the height of the triangle. But h is equal to the ordinate of point A, i.e. h=b sin C. Therefore, . The theorem has been proven.

Theorem 4.

The area of a trapezoid is equal to the product of half the sum of its bases and its height(Fig. 4.).

Figure 4.

Proof.

Let ABCD be the given trapezoid (Fig. 4.1.).

Figure 4.1.

The diagonal AC of a trapezoid divides it into two triangles: ABC and CDA.

Therefore, the area of the trapezoid is equal to the sum of the areas of these triangles.

The area of triangle ACD is equal to the area of triangle ABC. The heights AF and CE of these triangles are equal to the distance h between parallel lines BC and AD, i.e. height of the trapezoid. Hence, . The theorem has been proven.

The areas of figures are of great importance in geometry, as in science. After all, area is one of the most important quantities in geometry. Without knowledge of areas, it is impossible to solve many geometric problems, prove theorems, and justify axioms. The areas of figures were of great importance many centuries ago, but have not lost their importance in the modern world. Area concepts are used in many professions. They are used in construction, design and many other types of human activity. From this we can conclude that without the development of geometry, in particular the concepts of areas, humanity would not have been able to make such a big breakthrough in the field of science and technology.

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 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,

Area: Area is a quantity that measures the size of a surface. In mathematics, the area of a figure is a geometric concept, the size of a flat figure. Surface area is a numerical characteristic of a surface. Square in architecture, open... ... Wikipedia

Square- This term has other meanings, see Area (meanings). Area Dimension L² SI units m² ... Wikipedia

Area of a triangle- Standard notation A triangle is the simplest polygon having 3 vertices (angles) and 3 sides; part of the plane bounded by three points that do not lie on the same line and three segments connecting these points in pairs. Vertices of a triangle ... Wikipedia

Lenin Square (Petrozavodsk)- Lenin Square Petrozavodsk ... Wikipedia

Area (in geometry)- Area, one of the main quantities associated with geometric shapes. In the simplest cases, it is measured by the number of unit squares filling a flat figure, that is, squares with a side equal to one unit of length. Calculation of P. was already in ancient times... ...

SQUARE- one of the quantitative characteristics of flat geometric figures and surfaces. The area of a rectangle is equal to the product of the lengths of two adjacent sides. The area of a stepped figure (i.e. one that can be divided into several adjacent... ... Big Encyclopedic Dictionary

AREA (in geometry)- AREA, one of the quantitative characteristics of flat geometric shapes and surfaces. The area of a rectangle is equal to the product of the lengths of two adjacent sides. The area of a stepped figure (i.e. one that can be divided into several... ... encyclopedic Dictionary

SQUARE- AREA, squares, prev. about area and (obsolete) on area, plural. and areas, women. (book). 1. Part of a plane bounded by a broken or curved line (geom.). Area of a rectangle. Area of a curved figure. 2. only units. Space,… … Ushakov's Explanatory Dictionary

Area (architect.)- Square, an open, architecturally organized space, framed by any buildings, structures or green spaces, included in the system of other urban spaces. The predecessors of urban palaces were the ceremonial courtyards of palaces and... Great Soviet Encyclopedia

Memory Square (Tyumen)- Memory Square Tyumen General information ... Wikipedia

Books

Figures in mathematics, physics and nature. Squares, Triangles and Circles, Catherine Sheldrick-Ross. About the book Features of the book More than 75 unusual master classes will help turn the study of geometry into an exciting game The book describes the main figures in as much detail as possible: squares, circles and... Buy for 1206 rubles
Figures in mathematics, physics and nature Squares, triangles and circles, Sheldrick-Ross K.. More than 75 unusual master classes will help turn the study of geometry into an exciting game. The book describes the main figures in as much detail as possible: squares, circles, triangles. The book will teach...

Area formula is necessary to determine the area of a figure, which is a real-valued function defined on a certain class of figures of the Euclidean plane and satisfying 4 conditions:

Positivity - Area cannot be less than zero;
Normalization - a square with side unit has area 1;
Congruence - congruent figures have equal area;
Additivity - the area of the union of 2 figures without common internal points is equal to the sum of the areas of these figures.

Formulas for the area of geometric figures.

Geometric figure	Formula	Drawing
The result of adding the distances between the midpoints of opposite sides of a convex quadrilateral will be equal to its semi-perimeter.
Circle sector. The area of a sector of a circle is equal to the product of its arc and half its radius.
Circle segment. To obtain the area of segment ASB, it is enough to subtract the area of triangle AOB from the area of sector AOB.	S = 1 / 2 R(s - AC)
The area of the ellipse is equal to the product of the lengths of the major and minor semi-axes of the ellipse and the number pi.
Ellipse. Another option for calculating the area of an ellipse is through two of its radii.
Triangle. Through the base and height. Formula for the area of a circle using its radius and diameter.
Square . Through his side. The area of a square is equal to the square of the length of its side.
Square. Through its diagonals. The area of a square is equal to half the square of the length of its diagonal.
Regular polygon. To determine the area of a regular polygon, it is necessary to divide it into equal triangles that would have a common vertex at the center of the inscribed circle.	S= r p = 1/2 r n a

Definite integral. How to calculate the area of a figure

Let's move on to consider applications of integral calculus. In this lesson we will analyze the typical and most common task – how to use a definite integral to calculate the area of a plane figure. Finally, those who are looking for meaning in higher mathematics - may they find it. You never know. In real life, you will have to approximate a dacha plot using elementary functions and find its area using a definite integral.

To successfully master the material, you must:

1) Understand the indefinite integral at least at an intermediate level. Thus, dummies should first read the lesson Not.

2) Be able to apply the Newton-Leibniz formula and calculate the definite integral. You can establish warm friendly relations with certain integrals on the page Definite integral. Examples of solutions.

In fact, in order to find the area of a figure, you don’t need that much knowledge of the indefinite and definite integral. The task “calculate the area using a definite integral” always involves constructing a drawing, so your knowledge and drawing skills will be a much more pressing issue. In this regard, it is useful to refresh your memory of the graphs of basic elementary functions, and, at a minimum, to be able to construct a straight line, parabola and hyperbola. This can be done (for many, it is necessary) with the help of methodological material and an article on geometric transformations of graphs.

Actually, everyone has been familiar with the task of finding the area using a definite integral since school, and we will not go much further than the school curriculum. This article might not have existed at all, but the fact is that the problem occurs in 99 cases out of 100, when a student suffers from a hated school and enthusiastically masters a course in higher mathematics.

The materials of this workshop are presented simply, in detail and with a minimum of theory.

Let's start with a curved trapezoid.

Curvilinear trapezoid is a flat figure bounded by an axis, straight lines, and the graph of a function continuous on an interval that does not change sign on this interval. Let this figure be located not less x-axis:

Then the area of a curvilinear trapezoid is numerically equal to a definite integral. Any definite integral (that exists) has a very good geometric meaning. At the lesson Definite integral. Examples of solutions I said that a definite integral is a number. And now it’s time to state another useful fact. From the point of view of geometry, the definite integral is AREA.

That is, the definite integral (if it exists) geometrically corresponds to the area of a certain figure. For example, consider the definite integral. The integrand defines a curve on the plane located above the axis (those who wish can make a drawing), and the definite integral itself is numerically equal to the area of the corresponding curvilinear trapezoid.

Example 1

This is a typical assignment statement. The first and most important point in the decision is the construction of a drawing. Moreover, the drawing must be constructed RIGHT.

When constructing a drawing, I recommend the following order: at first it is better to construct all straight lines (if they exist) and only Then– parabolas, hyperbolas, graphs of other functions. It is more profitable to build graphs of functions point by point, the point-by-point construction technique can be found in the reference material Graphs and properties of elementary functions. There you can also find very useful material for our lesson - how to quickly build a parabola.

In this problem, the solution might look like this.
Let's draw the drawing (note that the equation defines the axis):

I will not shade the curved trapezoid; it is obvious here what area we are talking about. The solution continues like this:

On the segment, the graph of the function is located above the axis, That's why:

Answer:

Who has difficulties with calculating the definite integral and applying the Newton-Leibniz formula , refer to the lecture Definite integral. Examples of solutions.

After the task is completed, it is always useful to look at the drawing and figure out whether the answer is real. In this case, we count the number of cells in the drawing “by eye” - well, there will be about 9, it seems to be true. It is absolutely clear that if we got, say, the answer: 20 square units, then it is obvious that a mistake was made somewhere - 20 cells obviously do not fit into the figure in question, at most a dozen. If the answer is negative, then the task was also solved incorrectly.

Example 2

Calculate the area of a figure bounded by lines , , and axis

This is an example for you to solve on your own. Full solution and answer at the end of the lesson.

What to do if the curved trapezoid is located under the axle?

Example 3

Calculate the area of the figure bounded by lines and coordinate axes.

Solution: Let's make a drawing:

If a curved trapezoid is located under the axle(or at least not higher given axis), then its area can be found using the formula:
In this case:

Attention! The two types of tasks should not be confused:

1) If you are asked to solve simply a definite integral without any geometric meaning, then it may be negative.

2) If you are asked to find the area of a figure using a definite integral, then the area is always positive! That is why the minus appears in the formula just discussed.

In practice, most often the figure is located in both the upper and lower half-plane, and therefore, from the simplest school problems we move on to more meaningful examples.

Example 4

Find the area of a plane figure bounded by the lines , .

Solution: First you need to complete the drawing. Generally speaking, when constructing a drawing in area problems, we are most interested in the points of intersection of lines. Let's find the intersection points of the parabola and the straight line. This can be done in two ways. The first method is analytical. We solve the equation:

This means that the lower limit of integration is , the upper limit of integration is .
If possible, it is better not to use this method..

It is much more profitable and faster to construct lines point by point, and the limits of integration become clear “by themselves.” The point-by-point construction technique for various graphs is discussed in detail in the help Graphs and properties of elementary functions. Nevertheless, the analytical method of finding limits still sometimes has to be used if, for example, the graph is large enough, or the detailed construction did not reveal the limits of integration (they can be fractional or irrational). And we will also consider such an example.

Let's return to our task: it is more rational to first construct a straight line and only then a parabola. Let's make the drawing:

I repeat that when constructing pointwise, the limits of integration are most often found out “automatically”.

And now the working formula: If there is some continuous function on the segment greater than or equal to some continuous function , then the area of the figure bounded by the graphs of these functions and the lines , , can be found using the formula:

Here you no longer need to think about where the figure is located - above the axis or below the axis, and, roughly speaking, it matters which graph is HIGHER(relative to another graph), and which one is BELOW.

In the example under consideration, it is obvious that on the segment the parabola is located above the straight line, and therefore it is necessary to subtract from

The completed solution might look like this:

The desired figure is limited by a parabola above and a straight line below.
On the segment, according to the corresponding formula:

Answer:

In fact, the school formula for the area of a curvilinear trapezoid in the lower half-plane (see simple example No. 3) is a special case of the formula . Since the axis is specified by the equation, and the graph of the function is located not higher axes, then

And now a couple of examples for your own solution

Example 5

Example 6

Find the area of the figure bounded by the lines , .

When solving problems involving calculating area using a definite integral, a funny incident sometimes happens. The drawing was done correctly, the calculations were correct, but due to carelessness... the area of the wrong figure was found, this is exactly how your humble servant screwed up several times. Here is a real life case:

Example 7

Calculate the area of the figure bounded by the lines , , , .

Solution: First, let's make a drawing:

...Eh, the drawing came out crap, but everything seems to be legible.

The figure whose area we need to find is shaded blue(look carefully at the condition - how the figure is limited!). But in practice, due to inattention, a “glitch” often occurs that you need to find the area of a figure that is shaded in green!

This example is also useful in that it calculates the area of a figure using two definite integrals. Really:

1) On the segment above the axis there is a graph of a straight line;

2) On the segment above the axis there is a graph of a hyperbola.

It is quite obvious that the areas can (and should) be added, therefore:

Answer:

Let's move on to another meaningful task.

Example 8

Calculate the area of a figure bounded by lines,
Let’s present the equations in “school” form and make a point-by-point drawing:

From the drawing it is clear that our upper limit is “good”: .
But what is the lower limit?! It is clear that this is not an integer, but what is it? May be ? But where is the guarantee that the drawing is made with perfect accuracy, it may well turn out that... Or the root. What if we built the graph incorrectly?

In such cases, you have to spend additional time and clarify the limits of integration analytically.

Let's find the intersection points of a straight line and a parabola.
To do this, we solve the equation:

,

Really, .

The further solution is trivial, the main thing is not to get confused in substitutions and signs; the calculations here are not the simplest.

On the segment , according to the corresponding formula:

Answer:

Well, to conclude the lesson, let’s look at two more difficult tasks.

Example 9

Calculate the area of the figure bounded by the lines , ,

Solution: Let's depict this figure in the drawing.

Damn, I forgot to sign the schedule, and, sorry, I didn’t want to redo the picture. Not a drawing day, in short, today is the day =)

For point-by-point construction, it is necessary to know the appearance of a sinusoid (and in general it is useful to know graphs of all elementary functions), as well as some sine values, they can be found in trigonometric table. In some cases (as in this case), it is possible to construct a schematic drawing, on which the graphs and limits of integration should be fundamentally correctly displayed.

There are no problems with the limits of integration here; they follow directly from the condition: “x” changes from zero to “pi”. Let's make a further decision:

On the segment, the graph of the function is located above the axis, therefore:

Triangle area formulas

Square area formulas

Rectangle area formula

Parallelogram area formulas

Formulas for the area of ​​a rhombus

Trapezoid area formulas

Books

Definite integral. How to calculate the area of ​​a figure

Formulas for the area of a rhombus

Definite integral. How to calculate the area of a figure