Area of different figures. How to calculate the area of a figure

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 to 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 a factor of 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.

If you are planning to do the renovation yourself, then you will need to make an estimate for construction and finishing materials. To do this, you will need to calculate the area of the room in which you plan to carry out renovation work. The main assistant in this is a specially developed formula. The area of the room, namely its calculation, will allow you to save a lot of money on building materials and direct the freed-up financial resources in a more appropriate direction.

Geometric shape of the room

The formula for calculating the area of a room directly depends on its shape. The most typical for domestic buildings 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 projections.

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, just remember your school geometry lessons. Therefore, it should not be difficult for you to determine the area of the 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 agree, 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 reserve.

A room with a complex configuration

If your room does not fit 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 roughly divide the area of a room with this characteristic into rectangular elements and make calculations using the standard method. If you do not have this opportunity, then use the following methods:

Formula for finding the area of a circle:

S room=π*R 2, where

R is the radius of the room.

Formula for finding the area of a triangle:

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

P is the semi-perimeter of the triangle.

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

Hence P=A+B+C/2

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

Area of the room with projections and niches

Often the walls are decorated with decorative elements in the form of various niches or projections. 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 a flat section of the wall is found, and then the area of the niche or protrusion is added to it.

The area of the wall is found by the formula:

S walls = 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 based on the principle of finding the area of walls complicated by niches and projections.

However, if your ceiling design has arched and wave-like elements, then it is more appropriate to determine its area using the floor area. To do this you need:

Find the dimensions of all straight sections of 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 general

room area

Clear the room of unnecessary things. During the measurement process, you will need free access to all areas of your room, so you need to get rid of anything that might interfere with this.
Visually divide the room into regular and irregular shaped areas. If your room has a strictly square or rectangular shape, then you can skip this step.
Make a random layout of the room. This drawing is needed so that all the data is always at hand. 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 use it, make sure that 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 areas of individual sections of the room. Those. S total = S walls+S floor+S ceiling

The areas of geometric figures are numerical values characterizing their size in two-dimensional space. This value can be measured in system and non-system units. So, for example, a non-systemic unit of area is a hundredth, a hectare. This is the case if the surface being measured is a piece of land. The system unit of area is the square of length. In the SI system, the unit of flat surface area is the square meter. In the GHS, the unit of area is expressed as a square centimeter.

Geometry and area formulas are inextricably linked. This connection lies in the fact that the calculation of the areas of plane figures is based precisely on their application. For many figures, several options are derived from which their square dimensions are calculated. Based on the data from the problem statement, we can determine the simplest possible solution. This will facilitate the calculation and reduce the likelihood of calculation errors to a minimum. To do this, consider the main areas of figures in geometry.

Formulas for finding the area of any triangle are presented in several options:

1) The area of a triangle is calculated from the base a and height h. The base is considered to be the side of the figure on which the height is lowered. Then the area of the triangle is:

2) The area of a right triangle is calculated in the same way if the hypotenuse is considered the base. If we take the leg as the base, then the area of the right triangle will be equal to the product of the legs halved.

The formulas for calculating the area of any triangle do not end there. Another expression contains the sides a,b and the sinusoidal function of the angle γ between a and b. The sine value is found in the tables. You can also find it out using a calculator. Then the area of the triangle is:

Using this equality, you can also make sure that the area of a right triangle is determined through the lengths of the legs. Because angle γ is a right angle, so the area of a right triangle is calculated without multiplying by the sine function.

3) Consider a special case - a regular triangle, whose side a is known by condition or its length can be found when solving. Nothing more is known about the figure in the geometry problem. Then how to find the area under this condition? In this case, the formula for the area of a regular triangle is applied:

Rectangle

How to find the area of a rectangle and use the dimensions of the sides that have a common vertex? The expression for calculation is:

If you need to use the lengths of the diagonals to calculate the area of a rectangle, then you will need a function of the sine of the angle formed when they intersect. This formula for the area of a rectangle is:

Square

The area of a square is determined as the second power of the side length:

The proof follows from the definition that a square is a rectangle. All sides that form a square have the same dimensions. Therefore, calculating the area of such a rectangle comes down to multiplying one by the other, i.e., to the second power of the side. And the formula for calculating the area of a square will take the desired form.

The area of a square can be found in another way, for example, if you use the diagonal:

How to calculate the area of a figure that is formed by a part of a plane bounded by a circle? To calculate the area, the formulas are:

Parallelogram

For a parallelogram, the formula contains the linear dimensions of the side, height and mathematical operation - multiplication. If the height is unknown, then how to find the area of the parallelogram? There is another way to calculate. A certain value will be required, which will be taken by the trigonometric function of the angle formed by adjacent sides, as well as their length.

The formulas for the area of a parallelogram are:

Rhombus

How to find the area of a quadrilateral called a rhombus? The area of a rhombus is determined using simple math with diagonals. The proof is based on the fact that the diagonal segments in d1 and d2 intersect at right angles. The table of sines shows that for a right angle this function is equal to unity. Therefore, the area of a rhombus is calculated as follows:

The area of a rhombus can also be found in another way. This is also not difficult to prove, given that its sides are the same in length. Then substitute their product into a similar expression for a parallelogram. After all, a special case of this particular figure is a rhombus. Here γ is the interior angle of the rhombus. The area of a rhombus is determined as follows:

Trapezoid

How to find the area of a trapezoid through the bases (a and b), if the problem indicates their lengths? Here, without a known value of the height length h, it will not be possible to calculate the area of such a trapezoid. Because this value contains the expression for calculation:

The square size of a rectangular trapezoid can also be calculated in the same way. It is taken into account that in a rectangular trapezoid the concepts of height and side are combined. Therefore, for a rectangular trapezoid, you need to specify the length of the side side instead of the height.

Cylinder and parallelepiped

Let's consider what is needed to calculate the surface of the entire cylinder. The area of this figure is a pair of circles called bases and a side surface. The circles forming circles have radius lengths equal to r. For the area of a cylinder the following calculation takes place:

How to find the area of a parallelepiped that consists of three pairs of faces? Its measurements match the specific pair. Opposite faces have the same parameters. First, find S(1), S(2), S(3) - the square dimensions of the unequal faces. Then the surface area of the parallelepiped is:

Ring

Two circles with a common center form a ring. They also limit the area of the ring. In this case, both calculation formulas take into account the dimensions of each circle. The first of them, calculating the area of the ring, contains the larger R and smaller r radii. More often they are called external and internal. In the second expression, the ring area is calculated through the larger D and smaller d diameters. Thus, the area of the ring based on known radii is calculated as follows:

The area of the ring, using the lengths of the diameters, is determined as follows:

Polygon

How to find the area of a polygon whose shape is not regular? There is no general formula for the area of such figures. But if it is depicted on a coordinate plane, for example it could be checkered paper, then how to find the surface area in this case? Here they use a method that does not require approximately measuring the figure. They do this: if they find points that fall into the corner of the cell or have whole coordinates, then only them are taken into account. To then find out what the area is, use the formula proven by Peake. It is necessary to add the number of points located inside the broken line with half the points lying on it, and subtract one, i.e. it is calculated this way:

where B, G - the number of points located inside and on the entire broken line, respectively.

You can find over 10 formulas for calculating the area of a triangle on the Internet. Many of them are used in problems with known sides and angles of a triangle. However, there are a number of complex examples where, according to the conditions of the assignment, only one side and angles of a triangle are known, or the radius of a circumscribed or inscribed circle and one more characteristic. In such cases, a simple formula cannot be applied.

The formulas given below will allow you to solve 95 percent of problems in which you need to find the area of a triangle.
Let's move on to consider common area formulas.
Consider the triangle shown in the figure below

In the figure and below in the formulas, the classical designations of all its characteristics are introduced.
a,b,c – sides of the triangle,
R – radius of the circumscribed circle,
r – radius of the inscribed circle,
h[b],h[a],h[c] – heights drawn in accordance with sides a,b,c.
alpha, beta, hamma – angles near the vertices.

Basic formulas for the area of a triangle

1. The area is equal to half the product of the side of the triangle and the height lowered to this side. In the language of formulas, this definition can be written as follows

Thus, if the side and height are known, then every student will find the area.
By the way, from this formula one can derive one useful relationship between heights

2. If we take into account that the height of a triangle through the adjacent side is expressed by the dependence

Then the first area formula is followed by the second ones of the same type

Look carefully at the formulas - they are easy to remember, since the work involves two sides and the angle between them. If we correctly designate the sides and angles of the triangle (as in the figure above), we will get two sides a, b and the angle is connected to the third With (hamma).

3. For the angles of a triangle, the relation is true

The dependence allows you to use the following formulas for the area of a triangle in calculations:

Examples of this dependence are extremely rare, but you must remember that there is such a formula.

4. If the side and two adjacent angles are known, then the area is found by the formula

5. The formula for area in terms of side and cotangent of adjacent angles is as follows

By rearranging the indexes you can get dependencies for other parties.

6. The area formula below is used in problems when the vertices of a triangle are specified on the plane by coordinates. In this case, the area is equal to half the determinant taken modulo.

7. Heron's formula used in examples with known sides of a triangle.
First find the semi-perimeter of the triangle

And then determine the area using the formula

or

It is quite often used in the code of calculator programs.

8. If all the heights of the triangle are known, then the area is determined by the formula

It is difficult to calculate on a calculator, but in the MathCad, Mathematica, Maple packages the area is “time two”.

9. The following formulas use the known radii of inscribed and circumscribed circles.

In particular, if the radius and sides of the triangle, or its perimeter, are known, then the area is calculated according to the formula