For which shapes can you calculate the area? Finding the area of ​​a figure bounded by the lines y=f(x), x=g(y). Patch area

Knowledge of how to measure the Earth appeared in ancient times and gradually took shape in the science of geometry. WITH Greek language This word is translated as “land surveying”.

The measure of the extent of a flat section of the Earth in length and width is area. In mathematics, it is usually denoted by the Latin letter S (from the English “square” - “area”, “square”) or the Greek letter σ (sigma). S denotes the area of ​​a figure on a plane or the surface area of ​​a body, and σ is the cross-sectional area of ​​a wire in physics. These are the main symbols, although there may be others, for example, in the field of strength of materials, A is the cross-sectional area of ​​the profile.

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Calculation formulas

Knowing the areas of simple figures, you can find the parameters of more complex ones.. Ancient mathematicians developed formulas that can be used to easily calculate them. Such figures are triangle, quadrangle, polygon, circle.

To find the area of ​​a complex plane figure, it is broken down into many simple figures such as triangles, trapezoids or rectangles. Then mathematical methods derive a formula for the area of ​​this figure. A similar method is used not only in geometry, but also in mathematical analysis to calculate the areas of figures bounded by curves.

Triangle

Let's start with the simplest figure - a triangle. They are rectangular, isosceles and equilateral. Take any triangle ABC with sides AB=a, BC=b and AC=c (∆ ABC). To find its area, recall the well-known school course mathematics theorems of sines and cosines. Letting go of all calculations, we arrive at the following formulas:

  • S=√ - Heron’s formula, known to everyone, where p=(a+b+c)/2 is the semi-perimeter of the triangle;
  • S=a h/2, where h is the height lowered to side a;
  • S=a b (sin γ)/2, where γ is the angle between sides a and b;
  • S=a b/2, if ∆ ABC is rectangular (here a and b are legs);
  • S=b² (sin (2 β))/2, if ∆ ABC is isosceles (here b is one of the “hips”, β is the angle between the “hips” of the triangle);
  • S=a² √¾, if ∆ ABC is equilateral (here a is a side of the triangle).

Quadrangle

Let there be a quadrilateral ABCD with AB=a, BC=b, CD=c, AD=d. To find the area S of an arbitrary 4-gon, you need to divide it by the diagonal into two triangles, the areas of which S1 and S2 are in general not equal.

Then use the formulas to calculate them and add them, i.e. S=S1+S2. However, if a 4-gon belongs to a certain class, then its area can be found using previously known formulas:

  • S=(a+c) h/2=e h, if the tetragon is a trapezoid (here a and c are the bases, e is the midline of the trapezoid, h is the height lowered to one of the bases of the trapezoid;
  • S=a h=a b sin φ=d1 d2 (sin φ)/2, if ABCD is a parallelogram (here φ is the angle between sides a and b, h is the height dropped to side a, d1 and d2 are diagonals);
  • S=a b=d²/2, if ABCD is a rectangle (d is a diagonal);
  • S=a² sin φ=P² (sin φ)/16=d1 d2/2, if ABCD is a rhombus (a is the side of the rhombus, φ is one of its angles, P is the perimeter);
  • S=a²=P²/16=d²/2, if ABCD is a square.

Polygon

To find the area of ​​an n-gon, mathematicians break it down into the simplest equal figures - triangles, find the area of ​​each of them and then add them. But if the polygon belongs to the class of regular, then use the formula:

S=a n h/2=a² n/=P²/, where n is the number of vertices (or sides) of the polygon, a is the side of the n-gon, P is its perimeter, h is the apothem, i.e. a segment drawn from the center of the polygon to one of its sides at an angle of 90°.

Circle

A circle is a perfect polygon with an infinite number of sides. We need to calculate the limit of the expression on the right in the formula for the area of ​​a polygon with the number of sides n tending to infinity. In this case, the perimeter of the polygon will turn into the length of a circle of radius R, which will be the boundary of our circle, and will become equal to P=2 π R. Substitute this expression into the above formula. We will get:

S=(π² R² cos (180°/n))/(n sin (180°/n)).

Let's find the limit of this expression as n→∞. To do this, we take into account that lim (cos (180°/n)) for n→∞ is equal to cos 0°=1 (lim is the sign of the limit), and lim = lim for n→∞ is equal to 1/π (we translated degree measure into a radian, using the relation π rad=180°, and applied the first remarkable limit lim(sin x)/x=1 at x→∞). Substituting the obtained values ​​into the last expression for S, we arrive at the well-known formula:

S=π² R² 1 (1/π)=π R².

Units

Systemic and non-systemic units of measurement are used. System units belong to the SI (System International). This is a square meter (sq. meter, m²) and units derived from it: mm², cm², km².

In square millimeters (mm²), for example, the cross-sectional area of ​​wires in electrical engineering is measured, in square centimeters (cm²) - the cross-section of a beam in structural mechanics, in square meters(m²) - apartments or houses, in square kilometers(km²) - territories in geography.

However, sometimes non-systemic units of measurement are used, such as: weave, ar (a), hectare (ha) and acre (ac). Let us present the following relations:

  • 1 weave=1 a=100 m²=0.01 hectares;
  • 1 ha=100 a=100 acres=10000 m²=0.01 km²=2.471 ac;
  • 1 ac = 4046.856 m² = 40.47 a = 40.47 acres = 0.405 hectares.

There is an infinite number flat figures of various shapes, both regular and irregular. General property all figures - any of them has an area. The areas of figures are the dimensions of the part of the plane occupied by these figures, expressed in certain units. This quantity is always expressed positive number. The unit of measurement is the area of ​​a square whose side is equal to a unit of length (for example, one meter or one centimeter). The approximate area of ​​any figure can be calculated by multiplying the number of unit squares into which it is divided by the area of ​​one square.

Other definitions this concept look like this:

1. The areas of simple figures are scalar positive quantities that satisfy the conditions:

In equal figures - equal values areas;

If a figure is divided into parts (simple figures), then its area is the sum of the areas of these figures;

A square with a side of a unit of measurement serves as a unit of area.

2. Areas of figures complex shape(polygons) - positive quantities having the following properties:

Equal polygons have the same area sizes;

If a polygon is made up of several other polygons, its area is equal to the sum of the areas of the latter. This rule is valid for non-overlapping polygons.

It is accepted as an axiom that the areas of figures (polygons) are positive quantities.

The definition of the area of ​​a circle is given separately as the value to which the area of ​​a given circle inscribed in a circle tends - despite the fact that the number of its sides tends to infinity.

Areas of figures irregular shape(arbitrary figures) do not have a definition; only the methods for calculating them are determined.

Already in ancient times, calculating areas was an important practical task in determining the size of land plots. The rules for calculating areas over several hundred years were formulated by Greek scientists and set forth in Euclid's Elements as theorems. It is interesting that the rules for determining the areas of simple figures in them are the same as at present. Areas with a curved contour were calculated using the passage to the limit.

Calculating the areas of a simple rectangle or square), familiar to everyone from school, is quite simple. It is not even necessary to memorize the formulas for the areas of figures containing letter symbols. It is enough to remember a few simple rules:

2. The area of ​​a rectangle is calculated by multiplying its length by its width. It is necessary that the length and width be expressed in the same units of measurement.

3. We calculate the area of ​​a complex figure by dividing it into several simple ones and adding the resulting areas.

4. The diagonal of a rectangle divides it into two triangles whose areas are equal and equal to half of its area.

5. The area of ​​a triangle is calculated as half the product of its height and base.

6. The area of ​​a circle is equal to the product of the square of the radius and the well-known number “π”.

7. We calculate the area of ​​a parallelogram as the product of adjacent sides and the sine of the angle lying between them.

8. The area of ​​a rhombus is ½ the result of multiplying the diagonals by the sine of the interior angle.

9. Find the area of ​​a trapezoid by multiplying its height by its length midline, which is equal to the arithmetic mean of the bases. Another option for determining the area of ​​a trapezoid is to multiply its diagonals and the sine of the angle lying between them.

Children in primary school For clarity, tasks are often given: find the area of ​​a figure drawn on paper using a palette or a sheet of transparent paper, divided into squares. Such a sheet of paper is placed on the figure being measured, the number of complete cells (area units) that fit in its outline is counted, then the number of incomplete ones, which is divided in half.

To solve geometry problems, you need to know formulas - such as the area of ​​a triangle or the area of ​​a parallelogram - as well as simple techniques that we will cover.

First, let's learn the formulas for the areas of figures. We have specially collected them in a convenient table. Print, learn and apply!

Of course, not all geometry formulas are in our table. For example, to solve problems in geometry and stereometry in the second part profile Unified State Examination In mathematics, other formulas for the area of ​​a triangle are also used. We will definitely tell you about them.

But what if you need to find not the area of ​​a trapezoid or triangle, but the area of ​​some complex figure? There are universal ways! We will show them using examples from the FIPI task bank.

1. How to find the area of ​​a non-standard figure? For example, an arbitrary quadrilateral? A simple technique - let's divide this figure into those that we know everything about, and find its area - as the sum of the areas of these figures.

Divide this quadrilateral with a horizontal line into two triangles with common ground, equal to . The heights of these triangles are equal And . Then the area of ​​the quadrilateral is equal to the sum of the areas of the two triangles: .

Answer: .

2. In some cases, the area of ​​a figure can be represented as the difference of some areas.

It is not so easy to calculate what the base and height of this triangle are equal to! But we can say that its area is equal to the difference between the areas of a square with a side and three right triangles. Do you see them in the picture? We get: .

Answer: .

3. Sometimes in a task you need to find the area of ​​not the entire figure, but part of it. Usually we are talking about the area of ​​a sector - part of a circle. Find the area of ​​a sector of a circle of radius whose arc length is equal to .

In this picture we see part of a circle. The area of ​​the entire circle is equal to . It remains to find out which part of the circle is depicted. Since the length of the entire circle is equal (since ), and the length of the arc of a given sector is equal , therefore, the length of the arc is several times less than the length of the entire circle. The angle at which this arc rests is also a factor of less than a full circle (that is, degrees). This means that the area of ​​the sector will be several times smaller than the area of ​​the entire circle.

Instructions

It is convenient to act if your figure is a polygon. You can always break it down into a finite number, and you only need to remember one formula - the area of ​​a triangle. So, a triangle is half the product of the length of its side and the length of the altitude drawn to this very side. By summing up the areas of individual triangles into which a more complex triangle has been transformed by your will, you will find out the desired result.

It is more difficult to solve the problem of determining the area of ​​an arbitrary figure. Such a figure can have not only but also curved boundaries. There are ways to make an approximate calculation. Simple.

First, you can use a palette. This is an instrument made of transparent material with a grid of squares or triangles applied to its surface. known area. By placing the palette on top of the shape you're looking for area for, you recalculate the number of your units of measurement that overlap the image. Combine incompletely closed units of measurement with each other, completing them in your mind to complete ones. Next, by multiplying the area of ​​one palette shape by the number you calculated, you will find out the approximate area of ​​your arbitrary shape. It is clear that the more dense the grid is applied to your palette, the more accurate your result.

Secondly, you can outline the maximum number of triangles within the boundaries of an arbitrary figure for which you are determining the area. Determine the area of ​​each and add their areas. This will be a very rough result. If you wish, you can also separately determine the area of ​​the segments bounded by the arcs. To do this, imagine that the segment is part of a circle. Construct this circle, and then from its center draw radii to the edges of the arc. The segments form an angle α between themselves. The area of ​​the entire sector is determined by the formula π*R^2*α/360. For each smaller part of your figure, you determine the area and get the overall result by adding up the resulting values.

The third method is more difficult, but more accurate and for some, easier. The area of ​​any figure can be determined using integral calculus. The definite integral of a function shows the area from the graph of the function to the abscissa. The area enclosed between two graphs can be determined by subtracting definite integral, with a smaller value, from the integral within the same boundaries, but with a larger value. To use this method, it is convenient to transfer your arbitrary figure to a coordinate system and then define their functions and act using methods higher mathematics, which we will not delve into here and now.

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:

  1. Positivity - Area cannot be less than zero;
  2. Normalization - a square with side unit has area 1;
  3. Congruence - congruent figures have equal area;
  4. 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 opposite sides 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, you need to divide it into equal triangles, which would have a common vertex at the center of the inscribed circle.

S= r p = 1/2 r n a