How to find the area of geometric shapes. How to find the area of an irregular figure. Triangle. Through base and height

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,

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 of the profile Unified State Exam in mathematics, other formulas for the area of a triangle are 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 a common base 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 with a known area applied to its surface. 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 total 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 a certain integral, with a smaller value, from an 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 determine their functions and act using the methods of 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:

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

Each person has an idea of what the area of a room is, the area of a plot of land, the area of surface that needs to be painted. He also understands that if the plots of land are the same, then their areas are equal; that the area of the apartment consists of the area of the rooms and the area of its other premises.

This common idea of area is used when defining it in geometry, where they talk about the area of a figure. But geometric figures are arranged in different ways, and therefore, when they talk about area, they single out a certain class of figures.

For example, they consider the area of a polygon, the area of an arbitrary flat figure, the surface area of a polyhedron, etc. In our course we will only talk about the area of a polygon and an arbitrary flat figure.

Just as when considering the length of a segment and the magnitude of an angle, we will use the concept “consist of”, defining it as follows: figure F consists (composed) of figures F 1 and F 2, if it is their union and they have no common internal points.

In the same situation, we can say that the figure F is divided into figures F 1 and F 2. For example, about the figure F shown in Figure 2, a, we can say that it consists of figures F 1 and F 2, since they do not have common internal points. Figures F 1 and F 2 in Figure 2, b have common internal points, so it cannot be said that figure F consists of figures F 1 and F 2. If figure F consists of figures F 1 and F 2, then write: F=F 1 Å F 2.

Definition.The area of a figure is a positive quantity defined for each figure so that: 1) equal figures have equal areas; 2) if a figure consists of two parts, then its area is equal to the sum of the areas of these parts.

To measure the area of a figure, you need to have a unit of area. As a rule, such a unit is the area of a square with a side equal to a unit segment. Let us agree to denote the area of a unit square by the letter E, and the number that is obtained as a result of measuring the area of the figure - S(F). This number is called the numerical value of the area of the figure F with the selected unit of area E. It must satisfy the conditions:

1. The number S(F) is positive.

2. If the figures are equal, then the numerical values of their areas are equal.

3. If figure F consists of figures F 1 and F 2, then the numerical value of the area of the figure is equal to the sum of the numerical values of the areas of figures F 1 and F 2.

4. When replacing a unit of area, the numerical value of the area of a given figure F increases (decreases) by the same amount as the new unit is smaller (larger) than the old one.

5. The numerical value of the area of a unit square is taken equal to 1, i.e. S(F) = 1.

6. If figure F 1 is part of figure F 2, then the numerical value of the area of figure F 1 is not greater than the numerical value of the area of figure F 2, i.e. F 1 Ì F 2 Þ S (F 1) ≤ S (F 2) .

In geometry it has been proven that for polygons and arbitrary plane figures such a number always exists and is unique for each figure.

Figures whose areas are equal are called equal in size.

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How to calculate the area of a figure

In geometry problems, you often need to calculate the area of a flat figure. In stereometry tasks, the area of faces is traditionally calculated. It is often necessary to find the area of a figure in everyday life, for example, when calculating the number of building materials needed. There are special formulas for determining the area of the simplest figures. However, if a figure has a difficult shape, then calculating its area is sometimes not so easy.

You will need

calculator or computer, ruler, tape measure, protractor

Instructions

1. In order to calculate the area of a primitive figure, use the appropriate mathematical formulas: to calculate the area of a square, raise the length of its side to the second power: Pkv = c?, where: Pkv is the area of the square, c is the length of its side;

2. to find the area of a rectangle, multiply the lengths of its sides: Ppr = d * w, where: Ppr is the area of the rectangle, d and w are its length and width, respectively;

3. in order to find out the area of a parallelogram, multiply the length of each of its sides by the length of the height lowered to this side. If the lengths of adjacent sides of a parallelogram and the angle between them are known, then multiply the lengths of these sides by the sine of the angle between them: Ppar = C1 * B1 = C2 * B2 = C1 * C2 * sin?, where: Ppar is the area of the parallelogram, C1 and C2 are the lengths of the sides of the parallelogram, B1 and B2 are, respectively, the lengths of the heights lowered on them,? – the size of the angle between adjacent sides;

4. in order to find the area of a rhombus, multiply the length of the side by the length of the height or multiply the square of the side of the rhombus by the sine of any of its angles or multiply the lengths of its diagonals and divide the resulting product by two: Promb = C * B = C? *sin? = D1 * D2, where: Promb is the area of the rhombus, C is the length of the side, B is the length of the height, ? – the size of the angle between adjacent sides, D1 and D2 – the lengths of the diagonals of the rhombus;

5. to calculate the area of a triangle, multiply the length of the side by the length of the height and divide the resulting product by two, or multiply half the product of the lengths of 2 sides by the sine of the angle between them, or multiply the semi-perimeter of the triangle by the radius of the inscribed circle in the triangle, or take the square root of the product of the differences of the semi-perimeter of the triangle and each of its sides (Heron’s formula): Ptr = C * B / 2 = ? * C1 * C2 * sin? = p * p = ?(p*(p-C1)*(p-C2)*(p-C3)), where: C and B are the length of an arbitrary side and the height lowered onto it, C1, C2, C3 are the lengths sides of a triangle?

Area of figures

– the size of the angle between the sides (C1, C2), p – the semi-perimeter of the triangle: p = (C1+C2+C3)/2,p – the radius of the circle inscribed in the triangle;

7. to calculate the area of a circle, multiply the square of its radius by the number “pi”, approximately equal to 3.14: Pcr =? * р?, where: р – radius of the circle, ? – number “pi” (3.14).

8. To calculate the area of more complex figures, divide them into several non-overlapping primitive figures, find the area of each of them and add up the resulting results. Sometimes it is easier to calculate the area of a figure as the difference between the areas of 2 (or several) primitive figures.

Video on the topic

Area of a complex figure. 5th grade

Two figures are called equal if one of them can be superimposed on the other in such a way that these figures coincide. The areas of equal figures are equal. Their perimeters are also equal. Area of a Square To calculate the area of a square, you need to multiply its length by itself.

S = a aExample:SEKFM = EK EK

SEKFM = 3 3 = 9 cm2

The formula for the area of a square, knowing the definition of degree, can be written as follows:

S = a2Area of a rectangleTo calculate the area of a rectangle, you need to multiply its length by its width.

S = a bExample:SABCD = AB BC

SABCD = 3 7 = 21 cm2
You cannot calculate the perimeter or area if the length and width are expressed in different units of length. Be sure to check that both the length and width are expressed in the same units, that is, both in cm, m, etc. Area of complex figures The area of the entire figure is equal to the sum areas of its parts. Task: find the area of the garden plot. Since the figure in the figure is neither a square nor a rectangle, its area can be calculated using the rule above. Let's divide the figure into two rectangles, whose areas we can easily calculate using the well-known formula. SABCE = AB BC
SEFKL = 10 3 = 30 m2
SCDEF = FC CD
SCDEF = 7 5 = 35 m2

To find the area of the entire figure, add the areas of the found rectangles. S = SABCE + SEFKL
S = 30 + 35 = 65 m2

Answer: S = 65 m2 is the area of the garden plot. The property below may be useful to you when solving area problems. The diagonal of a rectangle divides the rectangle into two equal triangles. The area of any of these triangles is equal to half the area of the rectangle. Consider a rectangle: AC is the diagonal of rectangle ABCD.

Let's find the area of triangles ABC and ACD. First, find the area of the rectangle using the formula.SABCD = AB BC
SABCD = 5 4 = 20 cm2

S ABC = SABCD: 2

S ABC = 20: 2 = 10 cm2

How to find the area of ​​geometric shapes. How to find the area of ​​an irregular figure. Triangle. Through base and height

Triangle area formulas

Square area formulas

Rectangle area formula

Parallelogram area formulas

Formulas for the area of ​​a rhombus

Trapezoid area formulas

How to calculate the area of ​​a figure

Instructions

Area of ​​figures

Area of ​​a complex figure. 5th grade

How to find the area of geometric shapes. How to find the area of an irregular figure. Triangle. Through base and height

Formulas for the area of a rhombus

How to calculate the area of a figure

Area of figures

Area of a complex figure. 5th grade