# How to calculate the angle of inclination of the roof

Projects constructed suburban mansions can consider many requirements, wishes and even fads or "whims" of the owner of their respective owners. But always their "native" common feature &# 8212; without a solid roof never do any of their buildings. And in the foreground should go not only architectural delights the customer's specific requirements as to the structure element in this regard. This reliability and stability of the truss system and roofing, roof full performance of its direct purpose - protection against moisture penetration (and in some cases, in addition, even heat and sound insulation), if necessary - the functionality located directly below the roof space.

How to calculate the angle of inclination of the roof

Design of the roof structure - it is extremely responsible and quite difficult, especially for complex configuration. The most sensible thing would be to entrust this work to professionals who own methods of carrying out the necessary calculations and appropriate software for this purpose. However, the home owner may also be interested in some of the theoretical aspects. For example, it is important to know how to calculate the angle of the roof on their own, at least approximately &# 8212; to start.

This will give the opportunity to immediately estimate the opportunity to realize their "copyright prikidok" &# 8212; for matching your actual conditions of the region, on the "architecture" of the roof, on the planned roofing materials on the use of the attic. To some extent, the calculated angle of slope of the roof will make a preliminary calculation of parameters and the number of timber for the rafter system, the total area of ​​the roofing.

### In what quantities it is more convenient to measure the angle of the roof?

It would seem - a completely unnecessary question, since all at school know that the angle is measured in degrees. But clarity is still needed, because in the technical literature, and reference tables, and in the usual everyday life of some skilled craftsmen are not uncommon, and other units of measurement - percentages or relative aspect ratio.

And one more thing you need clarification &# 8212; that is taken as the angle of the roof?

What is meant by roof pitch?

The inclination angle - the angle formed by the intersection of two planes: the horizontal and the plane of the roof slope. In the figure it is shown by the letter of the Greek alphabet α.

We are interested acute angles (obtuse skates can not be simply defined) lies in the range from 0 to 90 °. Ramps steeper than 50 ÷ 60 ° in its "pure" form, are extremely rare, and is usually for decorative roof - in the construction of the pointed towers in the Gothic style. However, there are exceptions - such steep slopes may be the bottom row of the rafters of the roof mansard.

The lower roof rafters mansard may be located at a very high angle

Yet often we have to deal with slopes lying in the range from 0 to 45 °

With degrees clear - all probably represent a protractor with its divisions. A ka to be with a different unit?

Nothing too complicated.

The relative aspect ratio - is the most simplified fraction showing the ratio of the height of the lifting ramp (shown above designated Latin H) To the projection of the roof slope to the horizontal plane (the diagram - L).

L - it may be, depending on the design of the roof, half the span (at symmetric gable roof) completely span (if pent roof), or in complex configurations roof indeed linear section defined conducted to the horizontal projection. For example, in Scheme mansard roof portion of such a well is shown - the horizontal beam from the angle to the vertical upright, extending from the top of the lower rafter.

slope angle and recorded fraction, e.g. "13".

However, in practice it often happens that the use value of the slope angle in such a representation would be extremely inconvenient if, for example, the numbers in the fraction obtained non-circular and irreducible. For example, there is little to say inexperienced builder ratio 3: 11. In that case, it is possible to use another value measurement roof pitch - percent.

This value is very simple - you just need to find the result of dividing the already mentioned fractions and then multiply by 100. For example, in the above example  3: 11

3: 11 = 0,2727 × 100 = 27.27%

Thus, the value obtained slope of the roof slope, expressed as a percentage.

And what to do if you want to switch from degrees to percent, or vice versa?

You can remember this relationship. 100 % &# 8212; is the angle 45 degrees when the legs of the right triangle equal to each other, i.e. in this case the height of the ramp is equal to the length of its horizontal projection.

In this case, 45 ° / 100 = 0,45 ° = 27'. One deviation percentage is 27 angular minutes.

If you come from the other side, 100/45 ° = 2,22%. That is, we find that one degree - this is 2, 22% slope.

For ease of transfer values ​​from one to the other, you can use the table:

The value in gradusahZnachenie in% value in gradusahZnachenie in% value in gradusahZnachenie in%
12.22% 1635.55% 31 °68.88%
2 °4.44% 17 °37.77% 32 °71.11%
3 °6.66% 1840.00% 33 °73.33%
48.88% 1942.22% 34 °75.55%
511.11% 2044.44% 35 °77.77%
6 °13.33% 21 °46.66% 36 °80.00%
7 °15.55% 22 °48.88% 37 °82.22%
817.77% 23 °51.11% 38 °84.44%
920.00% 24 °53.33% 39 °86.66%
1022.22% 25 °55.55% 40 °88.88%
eleven24.44% 26 °57.77% 41 °91.11%
1226.66% 27 °60.00% 42 °93.33%
1328.88% 28 °62.22% 43 °95.55%
1431.11% 29 °64.44% 44 °97.77%
1533.33% thirty66.66% 45 °100.00%

For clarity would be helpful to bring a flow chart that shows the relationship very accessible of all these linear parameters with the angle of slope, and the magnitude of its dimension.

Scheme A. Interdependence units roof inclination angle and allowable types of roof

By this figure has yet to return, when will be considered types of roofing.

Even easier would be to calculate the slope and angle of slope. if we use the built-in calculator placed below:

### The dependence of the type of roofing on the steepness of slope

When planning the construction of your own home, the owner of the site probably already spends "estimates" and its head, and with family members - will look like their future housing. The roof in this matter, of course, is one of paramount importance. And here it is necessary to take into account the fact that not every roofing material can be used on different roof slope steepness. In order to avoid misunderstandings later is required in advance to provide this relationship.

roofs distribution chart of the steepness of the ramp

Roofs of the angle of slope can be divided into flat (gradient up to 5 °), with a small gradient (from 6 to 30 °) and krutouklonnye, respectively, with an angle of inclination exceeding 30 °.

Each of roof types have their advantages and disadvantages. For example, flat roofs have a minimum area, but require special sealing measures. On steep roofs are not late snow masses, but they are more susceptible to wind load due to its "sail". And roofing material - because of their own technological and operational features has certain restrictions on the use of rays with different slopes.

Referring to Figure examined previously (scheme A). Black circles with arcuate arrows and blue numerals indicate the scope of the various roofing (arrowhead indicates marginally allowable slope steepness value):

1 - a shingle, wood chips, natural shingles. In the same area is the use and still used in the southern edges of reed roofs.

2 - natural single-piece covered with roofing tiles, bitumen-polymer tiles, slate tiles.

3 - roller bitumen-based materials, at least four layers, with the outer pebbled recessed into the layer of molten mastic.

4 - see item 3, but for the reliability of the roof only three layers of roll material.

5 - similar to the above-described rolled materials (not less than three layers), but without an outer protective pebbled.

6 - roll roofing materials, affixed to the hot mastic not less than two layers. Metal, corrugated board.

7 - asbestos cement corrugated sheets (slate) uniform profile.

8 - coating clay roofing tiles

9 - asbestos cement sheets reinforced profile.

10 - a roofing steel sheet flare connections.

eleven - slate conventional coating profile.

Thus, if you wish to cover a specific roof with roofing material type, the slope angle of slope to be scheduled in said frame.

### The dependence of the ridge height of the angle of the roof

For those readers who can remember the course of high school trigonometry, this section may seem uninteresting. They can just skip it and go on. But forgotten is the need to refresh the knowledge of the interdependence of angles and sides in a right triangle.

What is it? In this case, the construction of the roof is always in the calculations are repelled from the right-angled triangle. Two of his leg - the length of the projection onto the horizontal plane ramp (length span, half of the span, etc. - depending on the type of roof) and the height of the ramp at its highest point (at the ridge or at the transition to the upper rafters - when calculating the lower trusses an attic roof). It is clear that there is one constant - a span length. But the height can be changed by varying the angle of inclination of the roof.

Two main dependence, expressed in terms of sine and tangent of the angle of slope are shown in the table. There are other dependencies (via the cosine or cotangent) but in this case we need only these two trigonometric functions.

Graphic skhemaOsnovnye trigonometric relations
H - ridge height
S - roof slope length
L - half of the span lengths (for symmetrical gable roof) or length of flight (at a pent roof)
α - roof pitch
tg α = H / LH = L × tg α
sin α = H / SS = H / sin α

Knowing these trigonometric identities, we can solve almost all the problems on the preliminary design of truss construction.

For clarity &# 8212; triangle attached to the roof

So, if you want to "dance" on the clearly defined lifting height ridge, the ratio tg α = H / L not difficult to determine the angle.

By dividing the obtained number in the table are tangent angle in degrees. Trigonometric functions are often laid in engineering calculators, they are mandatory in Exel tables (for those who know how to work with this handy app. However, the calculation is carried out there is not in degrees but in radians). But to our reader is not distracted by the necessary searches relevant tables give tangent value in the range from 1 to 80 °.

UgolZnachenie tangensaUgolZnachenie tangensaUgolZnachenie tangensaUgolZnachenie tangent
tg (1 °)0.01746tg (21 °)0.38386tg (41 °)0.86929tg (61 °)1.80405
tg (2 °)0.03492tg (22 °)0.40403tg (42 °)0.9004tg (62 °)1.88073
tg (3 °)0.05241tg (23 °)0.42447tg (43 °)0.93252tg (63 °)1.96261
tg (4 °)0.06993tg (24 °)0.44523tg (44 °)0.96569tg (64 °)2.0503
tg (5 °)0.08749tg (25 °)0.46631tg (45 °)1tg (65 °)2.14451
tg (6 °)0.1051tg (26 °)0.48773tg (46 °)1.03553tg (66 °)2.24604
tg (7 °)0.12278tg (27 °)0.50953tg (47 °)1.07237tg (67 °)2.35585
tg (8 °)0.14054tg (28 °)0.53171tg (48 °)1.11061tg (68 °)2.47509
tg (9 °)0.15838tg (29 °)0.55431tg (49 °)1.15037tg (69 °)2.60509
tg (10 °)0.17633tg (30 °)0.57735tg (50 °)1.19175tg (70 °)2.74748
tg (11 °)0.19438tg (31 °)0.60086tg (51 °)1.2349tg (71 °)2.90421
tg (12 °)0.21256tg (32 °)0.62487tg (52 °)1.27994tg (72 °)3.07768
tg (13 °)0.23087tg (33 °)0.64941tg (53 °)1.32704tg (73 °)3.27085
tg (14 °)0.24933tg (34 °)0.67451tg (54 °)1.37638tg (74 °)3.48741
tg (15 °)0.26795tg (35 °)0.70021tg (55 °)1.42815tg (75 °)3.73205
tg (16 °)0.28675tg (36 °)0.72654tg (56 °)1.48256tg (76 °)4.01078
tg (17 °)0.30573tg (37 °)0.75355tg (57 °)1.53986tg (77 °)4.33148
tg (18 °)0.32492tg (38 °)0.78129tg (58 °)1.60033tg (78 °)4.70463
tg (19 °)0.34433tg (39 °)0.80978tg (59 °)1.66428tg (79 °)5.14455
tg (20 °)0.36397tg (40 °)0.8391tg (60 °)1.73205tg (80 °)5.67128

If, on the contrary, when the basis is the angle of inclination of the roof, the ridge height is determined by the feedback arrangement to the formula:

H = L × tg α

Now, with the values ​​of two of the legs and the roof pitch, and is very easy to calculate the required length of the rafters from ridge to eaves. You can apply the Pythagorean theorem

S = √ (L² + H²)

Or, it's probably easier, as is already known angle value, use trigonometric relationship:

S = H / sin α

Meaning sines of the angles &# 8212; in the table below.

UgolZnachenie sinusaUgolZnachenie sinusaUgolZnachenie sinusaUgolZnachenie sine
sin (1 °)0.017452 sin (21 °)0.358368sin (41 °)0.656059sin (61 °)0.87462
sin (2 °)0.034899sin (22 °)0.374607sin (42 °)0.669131sin (62 °)0.882948
sin (3 °)0.052336sin (23 °)0.390731sin (43 °)0.681998sin (63 °)0.891007
sin (4 °)0.069756sin (24 °)0.406737sin (44 °)0.694658sin (64 °)0.898794
sin (5 °)0.087156sin (25 °)0.422618sin (45 °)0.707107sin (65 °)0.906308
sin (6 °)0.104528sin (26 °)0.438371sin (46 °)0.71934sin (66 °)0.913545
sin (7 °)0.121869sin (27 °)0.45399sin (47 °)0.731354sin (67 °)0.920505
sin (8 °)0.139173sin (28 °)0.469472sin (48 °)0.743145sin (68 °)0.927184
sin (9 °)0.156434sin (29 °)0.48481sin (49 °)0.75471sin (69 °)0.93358
sin (10 °)0.173648sin (30 °)0.5sin (50 °)0.766044sin (70 °)0.939693
sin (11 °)0.190809sin (31 °)0.515038sin (51 °)0.777146sin (71 °)0.945519
sin (12 °)0.207912sin (32 °)0.529919sin (52 °)0.788011sin (72 °)0.951057
sin (13 °)0.224951sin (33 °)0.544639sin (53 °)0.798636sin (73 °)0.956305
sin (14 °)0.241922sin (34 °)0.559193sin (54 °)0.809017sin (74 °)0.961262
sin (15 °)0.258819sin (35 °)0.573576sin (55 °)0.819152sin (75 °)0.965926
sin (16 °)0.275637sin (36 °)0.587785sin (56 °)0.829038sin (76 °)0.970296
sin (17 °)0.292372sin (37 °)0.601815sin (57 °)0.838671sin (77 °)0.97437
sin (18 °)0.309017sin (38 °)0.615661sin (58 °)0.848048sin (78 °)0.978148
sin (19 °)0.325568sin (39 °)0.62932sin (59 °)0.857167sin (79 °)0.981627
sin (20 °)0.34202sin (40 °)0.642788sin (60 °)0.866025sin (80 °)0.984808

Skilful use of trigonometric formulas allows, under normal spatial imagination and the ability to perform simple drawings and make calculations more complicated construction of roofs.

Based on the basic ratio, it is easy to divide into triangles and calculate the gambrel roof

For example, even such apparent "heaped up" hipped roof terrace or can be partitioned into plurality of triangles, and then sequentially count the necessary dimensions.

### Dependence of size of the room attic on the angle of the roof slopes

If the owners of the future house you plan to use the attic as a functional space, in other words - to make the attic, the determination of the angle of slope of the roof becomes quite practical significance.

The greater the slope angle &# 8212; the spacious attic

Many do not have to explain anything here - this scheme clearly shows that the smaller the angle of inclination, the tighter space in the attic.

To make it more understandable, it is better to perform a similar scheme in a certain scale. Here, for example, would look like an attic in the house with a wide gable of the 10 meters. Note that the ceiling height can not be less than 2 meters. (Frankly, not enough and two meters for residential pomescheniya- ceiling will inevitably "put pressure" on the person. Usually, from the height of at-least 2.5 meters).

to sample &# 8212; scaled circuit attic

Can lead already calculated the mean values ​​obtained in the attic room, depending on the angle of ordinary gable roof. In addition, the table shows the values ​​of the length of the rafters and squares roofing material with the 0.5 meters eaves of the roof.

The angle of slope kryshiVysota konkaDlina skataPoleznaya room attic area of ​​1 meter of length of the building (with a ceiling height of 2 m) Area roofing of the building 1 meter length
201.825.32no11.64
252.335.520.9212.03
thirty2.895.772.6112.55
353.506.103.8013.21
404.206.534.7514.05
455.007.075.5215.14
505.967.786.1616.56

Thus, the steeper the slope of the ramps, the spacious room. However, it immediately responds to a sharp increase in the height of truss construction, an increase in size, and therefore - and the weight of parts for its installation. Much more is required, and roofing material - coverage area is also growing rapidly. Plus, we can not forget about the increasing "sail" effect &# 8212; greater exposure to wind load. Types of external loads will be devoted to the last chapter of this publication.

for comparison &# 8212; mansard roof gives a useful gain in space, even with a reduced height

To a certain extent neutralize these negative effects, the designers and builders often use special design mansard roof - about it has already been mentioned in this article. It is more difficult in the calculation and production, but provides a significant gain in the resulting usable space attic room with a decrease in the overall height of the building.

### The dependence of the external loads on the angle of the roof

Another important applications of the calculated values ​​of the slope angle of the roof - the definition of the degree of its impact on the level of external loads falling on the roof structure.

There is an interesting relationship can be traced. You can pre-calculate all the parameters - the angles and linear dimensions, but always end up coming to the detailing. That is necessary to determine what material will be manufactured parts and assemblies truss system, what should be their cross-sectional area, the location of a step, the maximum length between adjacent points of support, methods of fastening elements to each other and to the supporting walls of the building and much more.

Here at the forefront of load experienced by the roof structure. In addition to its own weight, are of paramount importance external influences. If you do not take into calculation unusual for our edges seismic loads, the main focus should be on the snow and wind. The value of both - is directly related to the location of the roof angle to the horizon.

#### snow load

It is clear that the vast territory of the Russian Federation average statistical number of falls as snow precipitation varies considerably by region. According to the results of many years of observations and calculations, a map of the country, which indicated eight different zones on the level of snow load.

Map of the distribution zones in the territory of the Russian Federation on the snow load

The eighth and last zone - these are some sparsely populated areas of the Far East, and it can not be considered separately. The values ​​of other zones - are listed in the table

The RF zonal distribution of the average value in the snow nagruzkiZnachenie kPaZnachenie in kg / m²
I0.8 kPa80 kg / m
II1.2 kPa120 kg / m
III1.8 kPa180 kg / m
IV2.4 kPa240 kg / m
V3.2 kPa320 kg / m
VI4.0 kPa400 kg / m
VII4.8 kPa480 kg / m

Now, to calculate the specific load for the planned building, it is necessary to use the formula:

RSN = Rsn.t × μ

Rsn.t - the value that we found with the help of maps and tables;

Μ - correction factor which depends on the slope angle α

• at α from 0 before 25 ° &# 8212; μ = 1
• at α more 25 and up 60 ° &# 8212; μ = 0,7
• at α more 60 ° snow load not taken into account, because the snow does not stay on the plane of the roof ramps.

For example, a house built in Bashkiria. Planned rays of its roof - to 35 °.

Reading of the table - the zone V, the table value &# 8212; Rsn.t = 3.2 kPa

We find the final value RSN = 3.2 × 0,7 = 2,24 kPa

(If the value is to be in kilograms per square meter, use ratio

1 kPa ≈ 100 kg / m

In this case, a 224 kg / m².

#### wind load

With the wind load, everything is much more complicated. The fact that it could be different directions - the wind is able to exert pressure on the roof, pinning her to the ground, but at the same time there are aerodynamic 'lift' forces trying to tear the roof off the walls.

In addition, the wind load acts on different parts of the roof is uneven, so know only the average level of wind load - is not enough. Are taken into account the prevailing wind direction in the area ( "Roza Vetrov"), the degree of saturation of the site terrain obstacles to the spread of the wind, the height of the building and the surrounding buildings, and other criteria.

An exemplary procedure for calculating the wind load is as follows.

In the first place, by analogy with the previously conducted by calculations on the map is determined by the region of Russia and the corresponding zone.

Distribution zones in the territory of the Russian Federation on the level of wind pressure

Further, according to the table to determine the average value for a specific region of the wind pressure PBT

Regional distribution of the territory of the Russian Federation in terms of average wind nagruzkiIaIIIIIIIVVVIVII
Tabular value of wind pressure, kg / m ² (Pg)243242536784100120

Next, the calculation is carried out according to the following formula:

PB = PBT × k × c

PBT - a table value of wind pressure

k - coefficient taking into account the height of the building and the nature of the terrain around it. Define it on the table:

The height of the erected building (construction) (z) in the area azone BZona
not more than 5 m0.750.50.4
from 5 to 10 m1.00.650.4
from 10 to 20 m1.250.850.55
from 20 to 40 m1.51.10.8

The table shows the three different zones:

• Zone "A" &# 8212; Open the "bare" area, for example, steppe, desert, tundra and forest tundra, fully open wind impacts the coast of the seas and oceans, large lakes, rivers and reservoirs.
• Zone "B" &# 8212; territory of residential townships, towns, wooded and intersected terrain, obstacles to wind, natural or artificial, about 10 meters high.
• Zone "AT" &# 8212; the metropolitan area with dense buildings, with an average height of buildings of 25 meters and above.

The house is considered appropriate that zone if said characteristic features are located within a radius of no less than the building height h, multiplied by 30 (e.g., home area of ​​12 m radius must be not less than 360 m). At the height of buildings above 60 meters is accepted circle of 2000 m radius.

c - and here it is - the same factor that depends on the wind direction on the building and on the angle of the roof.

As already mentioned, depending on the direction of impact and features a roof wind can give multidirectional load vectors. The diagram below shows the impact zone of the windscreen, which is usually divided roof area.

Distribution of the building's roof into zones when calculating the wind load

Note - appears intermediate auxiliary quantity e. It was assumed to be either 2 × h, or b, Depending on the wind direction. In any case, the two values ​​taking that will be less.

Coefficient from for each of the zones is taken from the tables, which is considered the angle of the slope. If one portion provided both positive and negative values ​​of the coefficient, the two calculations are carried out and then the data are summarized.

Table coefficient "with "for wind directed to the slope of the roof

The angle of slope of the roof (α) FGHIJ
15- 0.9-0.8- 0.3-0.4-1.0
0.20.20.2
thirty-0.5-0.5-0.2-0.4-0.5
0.70.70.4
45 °0.70.70.6-0.2-0.3
60 °0.70.70.7-0.2-0.3
75 °0.80.80.8-0.2-0.3

Table coefficient "with "for wind directed at the part of the pediment

The angle of slope of the roof (α) FGHI
0 °-1.8-1.3-0.7-0.5
15-1.3-1.3-0.6-0.5
thirty-1.1-1.4-0.8-0.5
45 °-1.1-1.4-0.9-0.5
60 °-1.1-1.2-0.8-0.5
75 °-1.1-1.2-0.8-0.5

Now then, calculating wind loads, it will be possible to determine the total external force for each section of the roof.

Rsum = RSN + PB

The resulting value becomes a reference value for determining the parameters of truss system. In particular, in the table below, the values ​​can be found acceptable trusses free length between the support points, depending on the section bar, the distance between the rafters, the material grade (softwood) and, respectively, the total level of wind and snow loads.

Sort drevesinySechenie trusses (mm) distance between neighboring rafters (mm)
300400600300400600
total load (snow, wind +)1.0 kPa1.5 kPa
Wood is the highest grade40 × 893.222.922.552.812.552.23
140 × 405.064.604.024.424.023.54
184 × 506.656.055.285.815.284.61
235 × 508.507.726.747.426.745.89
286 × 5010.349.408.219.038.217.17
Class I or II40 × 893.112.832.472.722.472.16
140 × 404.904.453.894.283.893.40
184 × 506.445.855.115.625.114.41
235 × 508.227.476.507.186.525.39
286 × 5010.009.067.408.747.666.25
III grade40 × 893.062.782.312.672.391.95
140 × 404.674.043.303.953.422.79
184 × 505.684.924.024.804.163.40
235 × 506.956.024.915.875.084.15
286 × 508.066.986.706.815.904.82
total load (snow, wind +)2.0 kPa2.5 kPa
Wood is the highest grade40 × 894.023.653.193.733.392.96
140 × 405.284.804.194.904.453.89
184 × 506.746.135.356.265.694.97
235 × 508.217.466.527.626.925.90
286 × 502.472.241.962.292.081.82
Class I or II40 × 893.893.533.083.613.282.86
140 × 405.114.643.894.744.313.52
184 × 506.525.824.756.065.274.30
235 × 507.806.765.527.066.114.99
286 × 502.432.111.722.211.911.56
III grade40 × 893.483.012.463.152.732.23
140 × 404.233.672.993.833.322.71
184 × 505.184.483.664.684.063.31
235 × 506.015.204.255.434.713.84
286 × 506.525.824.756.065.274.30

It is understood that the calculation section trusses, and the step of setting the span (distance Mezhuyev support points) are taken of the total external pressure indicators for the most loaded areas of the roof. If you look at the circuit and the values ​​of the table of coefficients, it is - G and H.

To simplify the task of site visitors on the calculation of the total load, is placed below a calculator that calculates this parameter is for the most loaded areas.

#### A calculator for calculating the total, snow and wind load to determine the required cross-section of rafters

So, it is difficult to downplay the importance of a correct calculation of the angle of slope of the roof, the impact of this option on a number of important characteristics of the roof system, and the whole building. While holding this architectural calculations, of course, is increasingly the prerogative of professionals, the ability to navigate the basic concepts and perform basic calculations simple - it will be very useful for every literate home owner.

And at the end of the article - a video tutorial on the calculation of roof system conventional gable roof: