How to calculate the area of ​​a room

Someone, probably, will seem ridiculous so "children's" question - "how to calculate the area of ​​the room."But let's agree that, firstly, many of us have finished school for a very long time. Secondly, not all in everyday life are constantly faced with the need to perform geometric calculations of , and something could already be forgotten. And, thirdly, rooms and walls are not always and not everywhere have a rectangular shape, the area of ​​which is easy to calculate. Hence, sometimes will have to to be smart and apply more complicated methods of calculation.

And why in general you need to be able to determine the floor area, walls, ceiling of the room?

• These data are necessary for planning the repair in the apartment - you can calculate the number of required building materials.
• If the work is performed by hired masters, then they often require payment per square meter. In order not to become a victim of deception, to be able to control the integrity of employees and not to pay extra, it is necessary to be able to correctly determine
  volumes of work performed.
• The ability to count building schemes and determine the area may be useful when viewing residential or residential properties for sale or rental. By the way, the amount of payment for some utilities also depends on the size of the useful and total area.
• The correct selection of lighting devices directly depends on the area of ​​the room - account has special lighting standards.
• Knowing the area, it's easy to calculate and the volume of the room. And is the main param tr pr and selection of heating devices, climatic equipment, ventilation systems.

In a word, this skill will never be superfluous. And if the reader came to this page, means he needed to learn or refresh calculation algorithms in memory. Therefore , we start to consider, on a principle "from simple - to difficult".

Preparation for calculations

This stage, in fact, includes two main points. It is required to prepare the place for performing the measurement work, the necessary for this tool and accessories, and in fact, to conduct the measurements by transferring the results to the diagram or drawing .

• Ideally, it is best to carry out all measurements in a room completely freed from furniture and other items. When carrying out repairs, this usually happens. But if the measurements are carried out for other purposes, then no one will take out everything from the room, of course, no one will undertake.

Dimensions of the lengths of the of the room sides are usually held along the walls. Therefore, , it is necessary to try to release these areas, so that it is possible to extend the tape of roulette completely, from wall to wall .

• The simplest way to check the rectangularity of a room is to measure the diagonals. Therefore , you may need space in the center of the room. Or, it should be possible to move objects that may interfere with measurements.

Surely there will be ironic comments - does the author, giving such advice, never heard about modern electronic laser roulette? Of course, these modern instruments help not only to quickly and accurately measure linear and angular values ​​- many of them have built-in functions for performing the necessary calculations of .

have such a tool. However, let's "fall to the ground" - such tool is not yet available to many. Professionals, yes, it is difficult to manage now without it. However, in everyday life, most people rely on traditional methods of measurement. By the way, for measuring by laser roulette the free space anyway all is equal to create it is necessary - for unobstructed passage of the beam.

• So, we proceed from the fact that measurements are carried out with a regular tape measure. In addition, in addition to the notepad for records, may need :

- a long line for straight lines to be fixed( for example, a rule or a long construction level may come up);

- maximally large ( than larger than - the better) building angle;

- improvised compasses - it can be a piece of strong cord tied to some sharp pin that will play the role of the center;

- to measure the angle, you need either a special tool or an calculation using trigonometric functions. However, in those techniques that will be proposed, angle measurement does not become an obligatory procedure - the will cost the linear values.

- perhaps, on the floor it will be necessary to carry out some auxiliary markings. To make the not smudge the floor with chalk or marker, you can use a thin paint tape.

• All measurements are better done twice - so there will be less chance of making any unfortunate mistake due to inattention.
• The tape tape during the measurements should not sag, in the measured area should be a straight line, in the stretch .

• Attention on the scale! Do not confuse the system with metric and inch - foot, as many measuring instruments bear both scales.
• It will be much easier to work if you plan in advance a rough layout of the room on paper. It is not necessary for to draw it in this case of super-neatly , in scale, in compliance with drawing standards.

. It is simply much more convenient to immediately transfer measurement results to paper, for further calculations of .

When measuring distances, they will need to be recorded in meters, rounded to a centimeter, for example, 0, 82 m or 5, 38 m .Then all the results will immediately be obtained in the required square meters.

• Some formulas for calculating are rather bulky, containing tt rigonometric functions or calculating square roots, that is, it will not be possible to calculate the "column".Well, if you have the ability to carry out calculations of in Excel - all clearly, plus the whole process can be saved in a file. If not - means the calculator is used. It is convenient to use what is built into the Windows - , just do not forget to translate it from "normal" to "engineering" through the "View" menu or by simultaneously pressing the "Alt + 2" keys.

Now we are going to consider the possible configuration options for the room.

Determine the floor area of ​​the room

Rectangle

1. Probably the most common room configuration, especially in urban multi-storey buildings. And, accordingly, the most uncomplicated way calculation area.

The sides of the rectangle can be designated as A and B. Thus, the area will be:

S = And × In

The opposite sides of the rectangle are equal to each other, in pairs, that is, in fact, you can measure only two values.

But at the same time, you should not forget about the check - is it a rectangle? You can apply in each corner of the building corner, but not always this gives the a clear picture. The simpler will make measurements of the diagonals and compare the results. If they coincide or the difference is very small, then we can assume that this is really a rectangle.

In the example below, the area of ​​the room is:

S = A × In = 5.82 × 3.77 = 21.94 m²

2. The room can have ledges, niches, passages and , .rectangular shape. It's okay to split the total area into several rectangles on the diagram. It is clear that after calculating the areas by sections, all values ​​are summed up.

The diagram shows an example of a large room with a protruding section. On the diagram, the red dotted line shows the division of the room into two rectangular sections. It only remains to find the area of ​​each and summarize the of its :

Σ s = S 1 + S 2 = 8.26 × 3.73 + 2.63 × 1.50 = 30.81 + 3.95 = 34.75 m²

3. Sometimes it happens that the led , is directed towards the room and therefore "eats" part of the area.

Then with calculations you can go another by - find the area of ​​this "stolen" space and subtract its from the area of ​​the large rectangle.

Trapeze

This situation often happens in the corner rooms.

A characteristic feature of the quadrilateral , called a trapezoid - two sides of it are parallel to each other, and the other two can be located at an arbitrary angle.

In the given by the , in order to determine the area, you need to know the three parameters - the lengths of the two parallel sides and the height of the trapezoid, that is, the perpendicular distance between them.

S = ½( A + B ) × h

This formula is universal - nothing changes from the angles of the arrangement of the sides, and we do not need to know their meaning. But the height of h is better than to measure in several places - will be sure that the two sides are really parallel to each other.

Consider an example:

A typical trapezoidal room. We make measurements and apply the formula:

S = ½( A + B ) × h = 0.5 ×( 6.27 + 7.00) × 2.73 = 15.83 m²

Sometimes knowledge of the formula of the area of ​​the trapezoiduseful for complex, polygonal forms of the room. For example, the bottom pentagonal room is shown in the drawing. Its is most conveniently divided into two sections - a rectangle and a trapezoid( separated by a conditional red line).

Square of the rectangle:

S 1 = 6.33 × 3.95 = 25 m²

One side of the rectangle is the base of the trapezoid. A height of the trapezium is the difference in the lengths of the opposite sides of the room:

h = 7.71 - 6.33 = 1. 38 m

Find the area of ​​the trapezoid:

S 2 = 0.5 ×( 3.95 + 2.55) × 1.38 = 5. 71

Total,:

Σs = S 1 + S 2 = 25.0 + 5.71 = 30. 71

Calculator for calculating the area of ​​rectangular or trapezoidal spaces

The calculator below calculates the area of ​​both rectangular and trapezoidal rooms - the only difference is that the rectangular lengths are A1 and A2 will be the same.

If necessary, you can immediately amend the existing niches or ledges.

Triangles

So far, in all examples, some "correctness" of the - premises or rectangular, or at least have parallel walls, was observed. And what if the complex shape of the room does not allow to split its into such "correct" figures?

There is nothing to do, will have to to triangle a completely derived form, and then alternately calculate their area and sum.

The area of ​​the triangle is generally easy to calculate - it is necessary to multiply the length of its base by the height perpendicular to this base, from the opposite corner, and divide by two.

S = ½ × and × h

The difficulty is that it is precisely to repulse the perpendicular height in the conditions of measuring on the floor, at large distances, yes and without specialtools - not so easy. You can easily make a purely geometric error, which will pull the wrong result of the calculated area.

In this case, it is better to use the Geron formula for triangles. It, however, is somewhat cumbersome, but it allows you to accurately determine the area based solely on the lengths of the sides of the triangle. It looks like this:

S = √ p ×( p - a) × ( p - b) × ( p - with )

where:

- and , b and with - the lengths of the sides of the triangle;

- p - half of the perimeter of the triangle, i.e. p = ½( a + b + with )

We are looking at the example of a triangular room( sometimes there are also - inutility rooms, under the stairwells and )

So, the size of the corners is absolutely not interested in us - we measure only the lengths of all sides.

a = 6. 90 m

b = 3. 45 m

c = 6. 69 m

p = 0.5 ×( 6.90 + 3.45 + 6.69) = 8. 52

S = √ p ×( p - ( p -c ) =

= √ 8.52 ×( 8.52 - 6.90) ×( 8.52 - 3.45) ×( 8.52 -)6.69) =

= √ 8.52 × 1.62 × 5.07 × 1.83 = √ 128.06 = 11.32 m²

If you know how to calculate the area of ​​an arbitrary triangle on its sides, then do not represent t ore to determine the area of ​​almost any room with straight walls, regardless of the location of the sides. The example below shows the quadrangular room of an absolutely "irregular" shape. Split its into rectangles with a "makeweight" of triangles - there is no sense. Better already at once diagonally in two - we get two triangles.

The diagonal is shown in red, which becomes a common side for both triangles.

Armed with Heron's formula, calculate both areas and summarize them.

p 1 = 0.5 ×( 5.96 + 5.19 + 2.61) = 6. 88

S 1 = √ 6.88 ×( 6.88 - 5.96) ×( 6.88 - 5.19) ×( 6.88 - 2.61) = √45.67 =6.76 m²

p 2 = 0.5 ×( 5.96 + 6.63 + 4.34) = 8. 47

S 2 = √8.47 ×( 8.47 - 5.96) ×( 8.47 - 6.63) ×( 8.47 - 4.34) =√ 161.56 = 12.7 m²

Σs = S 1 + S 2 = 6.76 + 12.7 = 19.46 m²

At first glance, it may seem complicated and cumbersome, but in reality everything is simple. The main thing is to make careful measurements and not get confused in the calculations.

And to make the reader even easier, the calculator is placed below, instantly calculating the area of ​​the triangle along the lengths of its three sides. Note that the length values ​​must be entered in centimeters.

Calculate the area of ​​the triangular space

Curvilinear figures

It happens that there are areas in the room that have rounded shapes. So, we need to consider options for how to approach computations in this case. In addition, the skills of such calculations will be needed for the calculation of the wall area, on which there are arched door or window openings .

Too complex curvilinear figures, the area of ​​which can only be determined by integral calculus, will not be considered - these variants are extremely rare, and there is nothing to do here without special tools. But the figures based on the circle, it is possible to calculate. Most often in practice, there are semi-circle , a quarter circle or segment.

Circle and sector

The very area of ​​the circle is easy to find - you only need to know its radius( R).

S = π × R²

is needed. Now some examples of area determination with sectoral segments of the circle( half and quarter):

Needs to define the area of ​​a semicircular hall( or arches on a wall).The figure is fashionable divided into two - a rectangle( S 1) and a semicircle( S 2).

S 1 = 5.00 × 2.00 = 10.00 m²

S 2 = ½ × 3.14 × 2.50 ² = 9.8 m²

Σs = 10,00 + 9,8 = 19,8 m²

Another example when an arc connects two perpendicular walls, forming , so way , a quarter of a circle.

are connected to the arc. This room can be divided into three sections: two rectangles( their sides are highlighted with blue and yellow arrows), and a quarter of the circle with a radius marked with a green arrow.

The lower rectangle( blue):

S 1 = 3.67 ×( 3.0 - 1.5) = 5.51 m²

The upper rectangle( yellow ):

S 2 =( 3.67 - 1.5) × 1.5 = 3.25 m²

Quarter of the circle:

S 3 = ¼ × 3.14 × 1.5² = 1.76 m²

Total total area:

Σs = 5.51 + 3.25 + 1.76 = 10.52 m²

Segment

And now the task is somewhat more complicated than . Sometimes rounded arches or rooms take the form of not a sector, but a segment, that is, a part of the circle bounded by the arc of the and forming the arc of the arc of the with a chord( a segment connecting two points on the circle).

There are special formulas that allow to determine the area of ​​this figure on the basis of its not angular, but only linear dimensions.

The basic values ​​are the radius( R ) and the chord length ( C ) or height( H ).

The formula is the chord length:

S = R² × arcsin ( C / 2R) - 0.25 × C × √( 4R² - C² )

And if you repel the segment height, you can first determine the value of L :

L = R - H

And the formula takes the following form:

S = R² × arccos ( L / R) - L × √( R² - L² )

Let's look at an example.

You need to calculate the area of ​​such a room:

First of all, in this case it is necessary to find the center of the circle in order to determine with its radius. This can be done by experimenting the by the by moving the home-made compass( for example, from the cord) along the center line until a point is found. The distance from which the and up to the edges of the arc, and up to the of its center becomes equal. This distance is the radius.

Now it's easy to to measure and all other parameters. R = 2. 91 m , H = 1. 41 m , L = 1. 5 m .

As a result, we get two figures - a rectangle and a segment.

Area of ​​a rectangle:

S 1 = 5.00 × 2.00 = 10.00 m²

Find the segment area based on the radius and of the chord length ( in our case the chord length , quite obviously , iswidth of the room).

S 2 = 2.91² × arcsin ( 5 / ( 2 × 2.91) ) - 0.25 × 5.0 × √( 4 × 2.91² - 5.0²) =

= 5.0 1 m²

( For calculation on the calculator, be sure to installWhen calculating in Excel radians are set to default )

Just for the sake of interest - the same area, but through segment height:

S 2 = 2.91² × arccos ( 1.5 / 2.91) - 1.5 ×√( 2.91² - 1.5²) = 4.99 m²

The resulting difference of 0.02 m² is simply the result of rounding of long fractional numbers. Obviously, here it is quite possible to take an average value of 5.0 m²

Total, total area of ​​the room:

Σs = 10.00 + 5.00 = 15.00 m²

Determination of the areas of walls and ceiling

The floor in the room is not the only plane, the areawhich you need to know. Often, to calculate the number of necessary building or finishing materials, to determine volumes of work, you have to find the area of ​​walls and ceiling.

The laws of geometry are the same for all two-dimensional figures, so nothing new in their area of ​​mathematics is said. There are only some nuances:

Walls

It would seem, which is simpler: the wall area is its length multiplied by the height. Therefore, the total surface area of ​​the walls in the room can be found by making the perimeter of the room to the height.

Yes, this is true, but only for rooms where the ceiling height at the walls is the same at any point. And, for example, in mansard rooms the wall can be trapezoid or even triangular. Therefore , this is worth paying attention to. How to find the area of ​​a triangle or a trapezoid - we will not repeat.

Further - from the total area of ​​the wall it is necessary to subtract window or door openings .Nothing complicated - measurements are made on the outer surface of the boxes and frames, and the corresponding is calculated , depending on the geometric figure.

By the way, if the wall area is calculated in order to determine the number of standard wallpaper rolls for repair, then you can offer attention to the plate, which immediately prompts this number , based onfrom room perimeter and ceiling height:

Number of standard rolls of wallpaper 10,5 × 0, 52 m ( all door and window openings are included in the perimeter of the room).

Ceiling height( m)	Room perimeter( m)
9	10	12	13	14	15	16	17	19	20	21	22	23	25	26	27	28	30
2,15 ÷ 2,30	4	5	5	6	6	7	7	8	8	9	9	10	10	11	12	12	13	13
2,30 ÷ 2,45	5	5	6	6	7	7	8	8	9	9	10	10	11	11	12	13	13	14
2,45 ÷ 2,60	5	5	6	7	7	8	9	9	10	10	11	12	12	13	14	14	15	15
2,60 ÷ 2,75	5	5	6	7	7	8	9	9	10	10	11	12	12	13	14	14	15	15
2,75 ÷ 2, 90	6	6	7	7	8	9	9	10	10	11	12	12	13	14	14	15	15	16
2,90 ÷ 3,05	6	6	7	8	8	9	10	10	11	12	12	13	14	14	15	16	16	17
3,05 ÷ 3,20	6	7	8	8	9	10	10	11	12	13	13	14	15	16	16	17	18	19

Ceiling

In the overwhelming number of cases with the ceiling, the should not arise at all. If it is horizontal, then this is the exact projection of the floor, the area of ​​which has already been determined.

Attic, mansard rooms can be exceptions, since the plane of the ceiling there is at an angle to the floor, and sometimes the - is even broken, consisting of several surfaces intersecting each other.

It's okay. Do not go into trigonometry and calculate the angles between the floor, the walls and the ceiling. We just need to "virtually" break such complex surfaces into several flat shapes, measure their linear magnitudes. And the rest we already know how - formulas calculations will be exactly the same.

Finally, the completes the article - a video telling how an assistant can become a laser rangefinder( roulette).

Video: Area determination nonrectangular walls