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Areas of solid figures. How to calculate and label the area

Areas of solid figures. How to calculate and label the area

To solve problems in geometry, you need to know formulas - such as the area of a triangle or the area of \u200b\u200ba parallelogram - as well as simple tricks, which we will talk about.

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 exam 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 break this figure into those that we all know about, and find its area - as the sum of the areas of these figures.

Divide this quadrilateral by a horizontal line into two triangles with a common base equal to . The heights of these triangles are 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 \u200b\u200bthe figure can be represented as the difference of any areas.

It is not so easy to calculate what the base and height in 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-angled triangles. See them in the picture? We get: .

Answer: .

3. Sometimes in a task it is necessary to find the area not of the whole figure, but of its part. Usually we are talking about the area of \u200b\u200bthe sector - part of the circle. Find the area of \u200b\u200bthe sector of the circle of radius , whose arc length is equal to .

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

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

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

Formulas for the area of geometric shapes.

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 semiperimeter.
Circle sector. The area of a sector of a circle is equal to the product of its arc and half the radius.
circle segment. To get 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 an ellipse is equal to the product of the lengths of the major and minor semiaxes of the ellipse times pi.
Ellipse. Another option how to calculate the area of an ellipse is through its two radii.
Triangle. Through base and height. The formula for the area of a circle in terms of 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 diagonal. The area of a square is 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

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If you plan to make repairs yourself, then you will need to make an estimate for building and finishing materials. To do this, you will need to calculate the area of \u200b\u200bthe room in which you plan to carry out repairs. The main assistant in this is a specially designed formula. The area of \u200b\u200bthe room, namely its calculation, will allow you to save a lot of money on building materials and direct the released financial resources in a more necessary direction.

The geometric shape of the room

The formula for calculating the area of a room directly depends on its shape. The most typical for domestic structures are rectangular and square rooms. However, during redevelopment, the standard form may be distorted. The rooms are:

Rectangular.
Square.
Complex configuration (for example, round).
With niches and ledges.

Each of them has its own calculation features, but, as a rule, the same formula is used. The area of a room of any shape and size, one way or another, can be calculated.

Rectangular or square room

To calculate the area of a rectangular or square room, it is enough to remember school geometry lessons. Therefore, it should not be difficult for you to determine the area of \u200b\u200bthe room. The calculation formula looks like:

S rooms=A*B, where

A is the length of the room.

B is the width of the room.

To measure these values, you will need a regular tape measure. To get the most accurate calculations, it is worth measuring the wall on both sides. If the values do not converge, take the average of the resulting data as a basis. But remember that any calculations have their own errors, so the material should be purchased with a margin.

A room with a complex configuration

If your room does not fall under the definition of "typical", i.e. has the shape of a circle, triangle, polygon, then you may need a different formula for calculations. You can try to conditionally divide the area of \u200b\u200bthe room with such a characteristic into rectangular elements and make calculations in the standard way. If this is not possible for you, then use the following methods:

The formula for finding the area of a circle:

S room \u003d π * R 2, where

R is the radius of the room.

The formula for finding the area of a triangle is:

S room = √ (P (P - A) x (P - B) x (P - C)), where

P is the half-perimeter of the triangle.

A, B, C are the lengths of its sides.

Hence P \u003d A + B + C / 2

If in the process of calculating you have any difficulties, then it is better not to torture yourself and turn to professionals.

Room area with ledges and niches

Often the walls are decorated with decorative elements in the form of various niches or ledges. Also, their presence may be due to the need to hide some unaesthetic elements of your room. The presence of ledges or niches on your wall means that the calculation should be carried out in stages. Those. first, the area of \u200b\u200ba flat section of the wall is found, and then the area of \u200b\u200ba niche or ledge is added to it.

The area of the wall is found by the formula:

S walls \u003d P x C, where

P - perimeter

C - height

You also need to consider the presence of windows and doors. Their area must be subtracted from the resulting value.

Room with multi-level ceiling

A multi-level ceiling does not complicate the calculations as much as it seems at first glance. If it has a simple design, then calculations can be made on the principle of finding the area of \u200b\u200bwalls complicated by niches and ledges.

However, if the design of your ceiling has arcuate and undulating elements, then it is more appropriate to determine its area using the floor area. For this you need:

Find the dimensions of all straight sections of the walls.
Find the floor area.
Multiply the length and height of the vertical sections.
Sum the resulting value with the floor area.

Step-by-step instructions for determining the total

floor space

Free the room from unnecessary things. In the process of measuring, you will need free access to all areas of your room, so you need to get rid of everything that can interfere with this.
Visually divide the room into sections of regular and irregular shapes. If your room has a strictly square or rectangular shape, then this step can be skipped.
Make an arbitrary layout of the room. This drawing is needed so that all the data is always at your fingertips. Also, it will not give you the opportunity to get confused in numerous measurements.
Measurements must be taken several times. This is an important rule to avoid errors in calculations. Also if you are using make sure the beam lies flat on the wall surface.
Find the total area of the room. The formula for the total area of a room is to find the sum of all the areas of the individual sections of the room. Those. S total = S walls + S floors + S ceilings

Geometric area- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface bounded by a closed contour of this figure). The size of the area is expressed by the number of square units contained in it.

Triangle area formulas

Triangle area formula for 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
The formula for the area of a triangle given three sides and the radius of the circumscribed circle
The formula for the area of a triangle given three sides and the radius of an inscribed circle
Area of a triangle is equal to the product of the half-perimeter of the triangle and the radius of the inscribed circle.

Square area formulas

The formula for the area of a square given the length of a side
square area is equal to the square of its side length.
The formula for the area of a square given the length of the diagonal
square area equal to half the square of the length of its diagonal.
S= 1 2
2

Rectangle area formula

Rectangle area

Formulas for the area of a parallelogram

Parallelogram area formula for side length and height
Parallelogram area
The formula for the area of a parallelogram given two sides and the angle between them
Parallelogram area is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.
a b sinα

Formulas for the area of a rhombus

Rhombus area formula given side length and height
Rhombus area is equal to the product of the length of its side and the length of the height lowered to this side.
The formula for the area of a rhombus given the length of the side and the angle
Rhombus area 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.
The formula for the area of a rhombus from the lengths of its diagonals
Rhombus area is equal to half the product of the lengths of its diagonals.

Trapezium area formulas

Heron's formula for a trapezoid
Where S is the area of the trapezoid,
- the length of the bases of the trapezoid,
- the length of the sides of the trapezoid,