Building any roof is not as easy as it seems. And if you want it to be reliable, durable and not afraid of various loads, then first, at the design stage, you need to make a lot of calculations. And they will include not only the amount of materials used for installation, but also the determination of slope angles, slope areas, etc. How to calculate the roof slope angle correctly? It is on this value that the remaining parameters of this design will largely depend.
How to calculate the angle of a roof
Why is it important?
Design and construction of any roof is always a very important and responsible matter. Especially when it comes to the roof of a residential building or a roof with a complex shape. But even an ordinary lean-to, installed on a nondescript shed or garage, also needs preliminary calculations.
Roof project
If you do not determine in advance the angle of inclination of the roof, do not find out what the optimal height of the ridge should be, then there is a high risk of building a roof that will collapse after the first snowfall, or the entire finishing coating will be torn off even by a moderate wind.
Calculation of roof slope angle
Also, the angle of the roof will significantly affect the height of the ridge, the area and dimensions of the slopes. Depending on this, it will be possible to more accurately calculate the amount of materials required to create the rafter system and finishing materials.
The ridge is an important part of the rafter system
Prices for different types of roofing ridges
Roofing ridge
How designers choose the optimal angle
Calculations are made based on SNiP 2.01.07-85. The posted standards are used during calculations taking into account permanent, temporary and special loads and their various combinations.
SNiP 2.01.07-85. The PDF file will open in a new tab.
What loads are taken into account when determining the roof pitch angle?
Loads are divided into several categories depending on the duration of their impact: long-term, short-term and special.
Long-term (constant) loads on the rafter system
- . These include the weight of roofing materials, insulation, and wooden elements of the rafter structure. This category should include loads arising due to thermal expansion and changes in linear dimensions due to changes in the relative humidity of lumber. Standard temperature changes are determined using formulas separately for heated and unheated rooms. The weight of the snow cover is also considered a long-term load on the rafter system and is necessarily taken into account when determining the optimal angle of inclination of the rafter legs.
Load on the rafter system
Snow loads are considered long-term and are taken into account
Wind loads on the roof
Special stresses include earthquakes and natural disasters
When determining the roof slope angle, the maximum possible combination of loads is taken into account. Both of these parameters affect the thickness and length of the rafter legs. The calculation of the rafter system and the angle of inclination of the slopes is done according to limit states, taking into account all unfavorable factors.
The maximum deflections and movements of the rafter legs are regulated regardless of their linear dimensions and should not lead to partial depressurization of the roof. The following conditions apply to all types of roofs, regardless of the angle of inclination:
- The safe operation of buildings must be guaranteed;
- the integrity of the structure cannot be compromised even during short-term peak loads;
- the appearance of the roof should not change throughout the entire period of operation.
The rafter system must withstand peak loads without deformation
Moreover, each requirement must be fulfilled regardless of the others. The maximum deflection values of the rafters are limited taking into account the performance characteristics of the roofing materials. If the standard values do not have a noticeable effect on the appearance, then they are not adjusted.
Practical advice. It is much easier to ensure the integrity of the roof pie not by increasing the strength of the rafter system, but by using special structural compensators.
Shrinkage compensator, sliding support
Units
Remembering the geometry that everyone studied in school, it is safe to say that the angle of the roof is measured in degrees. However, in books on construction, as well as in various drawings, you can find another option - the angle is indicated as a percentage (here we mean the aspect ratio).
In general, the slope angle is the angle formed by two intersecting planes - the ceiling and the roof slope itself. It can only be sharp, that is, lie in the range of 0-90 degrees.
Small slope of the slopes
On a note! Very steep slopes, the angle of inclination of which is more than 50 degrees, are extremely rare in their pure form. Usually they are used only for decorative design of roofs; they can be present in attics.
As for measuring roof angles in degrees, everything is simple - everyone who studied geometry at school has this knowledge. It is enough to sketch out a diagram of the roof on paper and use a protractor to determine the angle.
The choice of roofing material depending on the slope of the roof
As for percentages, you need to know the height of the ridge and the width of the building. The first indicator is divided by the second, and the resulting value is multiplied by 100%. This way the percentage can be calculated.
On a note! At a percentage of 1, the typical degree of inclination is 2.22%. That is, a slope with an angle of 45 ordinary degrees is equal to 100%. And 1 percent is 27 arc minutes.
Table of values - degrees, minutes, percentages
What is percentage slope?
It is worth talking about the slope angle as a percentage. The very concept of “inclination angle as a percentage” is technically illiterate and is used only by those who do not build anything themselves.
Important. All rafter cuts are given in degrees; no one uses percentages when assembling the rafter system. Moreover, there are no measuring tools that measure angles as percentages. There are special tables for converting percentages to angles, but more on that a little later.
Let's consider these definitions in practice. Let's say the slope angle is 30%. What does it mean? This means that the height of the roof ridge is 30% of half the width of the building. For calculations we will use the same triangle.
Right triangle
The percentage of inclination is calculated using the formula (see figure below).
Incline percentage
Where:
- a – ridge height;
- b – half the width of the building.
It is very difficult to imagine what 30% is and what a roof with such a percentage looks like. In order to convert this value into degrees, you should use a special table. With its help, we find out that 30% means that the angle of inclination of the roof slope is approximately 16.5°. The fact is that for 16° the percentage is 28.7%, and for 17° this parameter is 30.5%. If the master knows that the slope angle is approximately 16.5°, then he can easily imagine the appearance and geometry of the roof, calculate the linear dimensions of the rafters, vertical roof supports, and the dimensions of the mauerlat. How are such calculations made if the percentage slope data is available?
Examples of using percentage slope
Calculation of the parameters of the rafter system using the percentage ratio of the height of the ridge and half the width of the building is done using a calculator.
Having the initial formula, further calculations are constructed using elementary arithmetic equations. First you need to transform the formula a little.
Initial formula
In this case, X is the percentage of the roof slope, as we agreed; for example, let’s take 30%. This value is known and specified during calculations.
For preliminary calculations, you should slightly transform the formula into this form (see the figure below).
Converted formula
Now we determine the values of a and b separately.
Value a
b value
Let us remind you that:
- a – height of the rafter system,
- b is half the width of the building, and
- X is the percentage of slope of the slope.
We know the percentage; for further calculations we will need to measure either the height of the rafter system or half the width of the building. Due to the fact that it is much easier to find out the second value, we will measure it.
Eg. The width of the building is 8 meters, respectively, half is equal to 4 meters (b = 4 m).
Find out the height of the rafter system (see figure below).
Find out the height of the rafter system
The height of the rafter system is 1.2 meters, we learned the angle of inclination from the table, it is approximately 16.5°.
Next, we need to calculate the length of the rafter leg without the eaves overhang. There are two ways to go.
First. Using the Pythagorean theorem, where c is the length of the rafter.
Pythagorean theorem
Accordingly, the value of c can be represented by the formula in the illustration below.
Formula for the length of the hypotenuse
Calculation example (see picture below).
Calculation example
Second. Using trigonometric functions. As we have already indicated above, the calculation of the length of the rafters can be done using the formula below.
Calculation of rafter length
We have all the data, the angle of inclination is 16.5°, half the width of the building is 4 m.
Calculation example
There are slight differences in the length of the rafters. This is because the angle of inclination was chosen approximately.
What factors influence the angle of inclination?
The angle of inclination of any roof is influenced by a very large number of factors, ranging from the wishes of the future owner of the house and ending with the region where the house will be located. When calculating, it is important to take into account all the subtleties, even those that at first glance seem insignificant. One day they may play their role. Determine the appropriate roof angle by knowing:
- types of materials from which the roof pie will be built, starting from the rafter system and ending with the external decoration;
- climate conditions in a given area (wind load, prevailing wind direction, amount of precipitation, etc.);
- the shape of the future building, its height, design;
- purpose of the building, options for using the attic space.
What affects the angle of the roof
In those regions where there is a strong wind load, it is recommended to build a roof with one slope and a small angle of inclination. Then, in a strong wind, the roof has a better chance of standing and not being torn off. If the region is characterized by a large amount of precipitation (snow or rain), then it is better to make the slope steeper - this will allow precipitation to roll/drain from the roof and not create additional load. The optimal slope of a pitched roof in windy regions varies between 9-20 degrees, and where there is a lot of precipitation - up to 60 degrees . An angle of 45 degrees will allow you to ignore the snow load as a whole, but in this case the wind pressure on the roof will be 5 times greater than on a roof with a slope of only 11 degrees.
On a note! The greater the roof slope parameters, the greater the amount of materials required to create it. The cost increases by at least 20%.
Sheathing frequency for different roofing materials
Why do you need to know the roof slope?
Without an accurate understanding of this issue, it is impossible to understand slopes in degrees and percentages. What is affected by the slope of the slopes?
| Roof parameter | Impact on roof construction |
Type of roofing material | Each roofing material has restrictions on the minimum angle of inclination. For example, on flat roofs (angle of inclination less than 10°) you can only use rolled roofing materials, for piece tiles the minimum inclination angle is 15°, etc. In addition, the amount of overlap between waterproofing and roofing materials depends on this parameter. |
| Wind and snow forces | The greater the angle of inclination, the less snow loads on the rafter system. With wind loads everything is more complicated. With small slopes of the slopes during strong gusts of wind, a lifting force acts on them; if the angle of inclination increases, an overturning force appears. Such changes in loads require a careful approach during the design and calculation of the rafter system. |
Consumption of materials | The greater the angle of inclination, the larger the area of the slopes covering the building plan. Accordingly, the consumption of roofing materials increases. For example, with a slope slope of 60°, the consumption of materials is twice as high as when covering a building with a flat pitched roof. |
As can be seen from the table, the angle of inclination of the roof slopes is one of the most important technical parameters of the rafter system of a house. They pay attention to it from the very beginning of the construction project.
Slope angles and roofing materials
Not only climatic conditions will have a significant impact on the shape and angle of the slopes. The materials used for construction, in particular roof coverings, also play an important role.
Installation of profiled sheets for roofing
Table. Optimal slope angles for roofs made of various materials.
| Type of material | Slope angle, degrees |
| Corrugated sheeting (metal) | 12 |
| Metal tiles | 14-25 |
| Ruberoid depending on the number of layers | 2-15 |
| Slate | 20-35 |
| Piece material such as roofing stones and tiles | 22-25 |
| Soft tiles | Minimum 11 |
On a note! The lower the roof slope, the smaller the pitch used when creating the sheathing.
We calculate the angle of inclination of the slopes
Prices for metal tiles
Metal tiles
The optimal angle of inclination of a gable roof
The angle of a regular gable roof lies within the range of 20°-45° , which corresponds to the spread of values of material properties and average climatic parameters.
It should be borne in mind that the recommended values may not be suitable for a given area or project, so each time you need to calculate the angle of a gable roof based on the specific data available.
The angle of inclination of a gable roof is an important indicator that affects the durability and integrity of the entire building, and it cannot be treated as a secondary factor.
Taking into account all possible loads, both permanent and one-time extreme ones, will help ensure the safety and comfort of your home .
More accurate values are selected based on factors such as:
- Purpose of the attic.
- Roof covering used.
- Climatic conditions.
The optimal angle of inclination of a gable roof
The height of the ridge also depends on the angle of the slope
When calculating any roof, a right-angled triangle is always taken as a reference point, where the legs are the height of the slope at the top point, that is, at the ridge or the transition of the lower part of the entire rafter system to the top (in the case of attic roofs), as well as the projection of the length of a particular slope on horizontal, which is represented by overlaps. There is only one constant value here - this is the length of the roof between the two walls, that is, the length of the span. The height of the ridge part will vary depending on the angle of inclination.
The height of the ridge may vary depending on the angle of inclination
Knowledge of formulas from trigonometry will help you design a roof: tgA = H/L, sinA = H/S, H = LxtgA, S = H/sinA, where A is the angle of the slope, H is the height of the roof to the ridge area, L is ½ of the entire length roof span (with a gable roof) or the entire length (in the case of a single-pitched roof), S – the length of the slope itself. For example, if the exact value of the height of the ridge part is known, then the angle of inclination is determined using the first formula. You can find the angle using the table of tangents. If the calculations are based on the roof angle, then the ridge height parameter can be found using the third formula. The length of the rafters, having the value of the angle of inclination and the parameters of the legs, can be calculated using the fourth formula.
Tangent table
How do the dimensions of the attic depend on the angle of inclination?
To make it possible to make useful use of the attic space, it is worth thinking about building an attic. And here the angle of inclination of the roof acquires the so-called applied significance. Depending on what this value is, the free space of the attic also depends. So, the smaller the angle of inclination, the less free space there will be in this part of the house.
Important! The ceiling height in the attic cannot be less than 2 m.
Roof angle
Thus, the attic should be built only with steep slopes. But in this case, certain problems arise: the dimensions of the roof increase, the height of the truss structure also increases, and there will be a need to design the mass of necessary small elements. Such a roof will “sail” more strongly and must be very durable in order to withstand various influences of external factors.
Calculation of the angle of inclination
In order to make correct calculations, you can use special online services. With their help, you can almost accurately select the necessary parameters and eliminate errors during construction.
It should be noted that when calculating the necessary parameters, the following values must be taken into account:
- rafter length;
- the height of the future roof ridge;
- span length of one roof slope;
- tangent of the angle of inclination.
Taking into account all these parameters, we can make an unambiguous conclusion that the lower the height of the future roof, the correspondingly shorter the length of the roof slope of a gable roof should be.
An incorrectly chosen angle of inclination, unfortunately, can incredibly shorten the maximum service life of the roof.
Snow load values
Russia is a huge country, and the climate in different parts of it can differ significantly from each other. Snow load indicators too. There are 8 main zones, divided by snow load intensity.
Map of distribution of zones by snow load
To calculate the snow load on the designed structure, use the formula Рсн = Рст.н x m, where Рсн.н. is an indicator determined using special tables, and m is the correction factor, which depends on the angle of the roof slope. It will be equal to 1 if the slope angle varies between 0-25 degrees, 0.7 - for slopes of 25-60 degrees. If the angle exceeds 60 degrees, then the snow load is not taken into account when designing the roof.
Zonal distribution by average snow load
Calculation of uneven load on a hipped roof
Prices for snow guards
Snow guard
Wind load values
Since the wind can change direction, identifying wind load will be much more difficult than snow load. Thanks to it, the roof can be pressed against the base, but can also be subject to a certain force that will tend to tear it off the house. Also, the wind affects the entire structure unevenly.
To carry out the necessary calculations, you will have to use only the prevailing wind direction in a given region, which is determined by the “wind rose”. Also, when making calculations, it is necessary to take into account the presence near buildings, mountains, forests and other elements that can not only change the direction of the wind, but also to some extent regulate its strength, protecting the structure from squalls.
Distribution of zones by wind load
According to the map, it is possible to identify the main wind characteristics prevailing in a certain area of the country. Next, the wind pressure Pwt (kg/m2) is determined. It will vary depending on the zone:
- Ia – 24;
- I – 32;
- II – 42;
- III – 53;
- IV – 67;
- V – 84;
- VI – 100;
- VII – 120.
Then the formula Pv = Pvt x K x C is used, where K is the value of the coefficient depending on the height of the building and terrain features, and C is the coefficient depending on the angle of inclination of the slope and wind direction.
Table. Determination of coefficient K.
| Building height, m | A | B | IN |
| Less than 5 | 0,75 | 0,5 | 0,4 |
| 5-10 | 1 | 0,65 | 0,4 |
| 10-20 | 1,25 | 0,85 | 0,55 |
| 20-40 | 1,5 | 1,1 | 0,8 |
Wind load
A, B, C are certain types of zones, A is an open bare area where the wind load will be maximum, zone B refers to small residential settlements with obstacles up to 10 m high, rough terrain or areas surrounded by forests, and B is a dense zone developments in cities where the height of buildings is 25 meters or more.
On a note! The value of the building height H, which is multiplied by 30, will help determine which zone to choose in each specific case. Thus, a suitable zone radius is obtained. For example, with a building height of 60 m, you should focus on a circle with a radius of 2 km.
Distribution of the roof of a building into zones when calculating wind load
According to the figure above, the e indicator is of great importance when determining the impact of wind on a certain section of the roof. It will be equal to 2xN or b (whichever is smaller is selected). Coefficient c is determined from the table, taking into account the angle of inclination of the roof slopes.
Table. Meaning C (pediment).
| Slope angle | G | F | I | H |
| 0 | -1,3 | -1,8 | -0,5 | -0,7 |
| 15 | -1,3 | -1,3 | -0,5 | -0,6 |
| 30 | -1,4 | -1,1 | -0,5 | -0,8 |
| 45 | -1,4 | -1,1 | -0,5 | -0,9 |
| 60 | -1,2 | -1,1 | -0,5 | -0,8 |
Table. Value C (slope).
| Slope angle | G | F | I | H | J |
| 15 | -0.8 or 0.2 | -0.9 or 0.2 | -0,4 | -0.3 or 0.2 | -1 |
| 30 | -0.5 or 0.7 | -0.5 or 0.7 | -0,4 | -0.2 or 0.4 | -0,5 |
| 45 | 0,7 | 0,7 | -0,2 | 0,6 | -0,3 |
| 60 | 0,7 | 0,7 | -0,2 | 0,7 | -0,3 |
The total force impact on each section of the roof is calculated by the formula: Рsum = Рсн + Рв. This indicator will become the starting point for calculating the rafters.
On a note! The easiest way to calculate the angle of the slope is not to do it yourself, but to use online calculators or computer programs.
How to calculate the roof slope angle: use a calculator
Projects of country mansions being built can take into account many requirements, wishes and even the whims or “whims” of their owners. But they are always “related” by a common feature - none of their buildings can ever do without a reliable roof. And in this matter, what should come to the fore is not so much the architectural delights of the customer, but rather the specific requirements for this element of the structure. This is the reliability and stability of the entire rafter system and roofing covering, the full performance by the roof of its direct purpose - protection from moisture penetration (and in some cases, in addition, thermal and sound insulation), and, if necessary, the functionality of the premises located directly under the roof.
How to calculate the angle of a roof
Designing a roof structure is an extremely responsible and quite difficult task, especially with complex configurations. It would be wisest to entrust this matter to professionals who know the methodology for carrying out the necessary calculations and the appropriate software for this. However, the home owner may also be interested in some theoretical points. For example, it is important to know how to calculate the angle of the roof yourself, at least approximately - for starters.
This will make it possible to immediately estimate the possibility of implementing your “author’s estimates” - according to the correspondence of the plan to the real conditions of the region, according to the “architecture” of the roof itself, according to the planned roofing material, according to the use of the attic space. To a certain extent, the calculated angle of the roof slope will help to make a preliminary calculation of the parameters and quantity of lumber for the rafter system, and the total area of the roof covering.
In what quantities is it more convenient to measure the angle of the roof slope?
It would seem a completely unnecessary question, since everyone knows from school that angle is measured in degrees. But clarity is still needed here, because in technical literature, in reference tables, and in the everyday life of some experienced craftsmen, other units of measurement are often found - percentages or relative aspect ratios.
And one more necessary clarification - what is taken as the angle of inclination of the roof?
What is meant by the angle of the roof?
The angle of inclination is 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 α of the Greek alphabet.
The acute angles that interest us (there cannot be obtuse angles simply by definition) lie in the range from 0 to 90°. Slopes steeper than 50 ÷ 60 ° in a “pure” form are extremely rare, and then, as a rule, for the decorative design of roofs - during the construction of pointed towers in the Gothic style. However, there is an exception - the slopes of the lower row of rafters of a mansard-type roof can be so steep.
The lower rafters of a mansard roof can be positioned at a very large angle
And yet, most often you have to deal with slopes lying in the range from 0 to 45°
With degrees it’s clear - everyone probably imagines a protractor with its divisions. What about other units of measurement?
Nothing complicated either.
The relative aspect ratio is the most simplified fraction showing the ratio of the height of the rise of the slope (in the figure above indicated by the Latin H ) to the projection of the roof slope onto the horizontal plane (in the diagram - L ).
L - this can be, depending on the roof design, half the span (with a symmetrical gable roof), the entire span (if the roof is single-pitched), or, with complex roof configurations, a truly linear section, determined by the projection drawn to the horizontal plane. For example, in the diagram of an attic roof such a section is clearly shown - along a horizontal beam from the very corner to a vertical post running from the top point of the lower rafter.
The slope angle is written as a fraction, for example “1 : 3».
However, in practice it often happens that using the slope angle in such a representation will be extremely inconvenient if, say, the numbers in the fraction are non-round and irreducible. 3:11 will have little meaning to an inexperienced builder . In this case, it is possible to use another value for measuring the roof slope - percentages.
Finding this value is extremely simple - you just need to find the result of dividing the already mentioned fraction, and then multiply it by 100. For example, in the example above 3 : 11
3 : 11 = 0,2727 × 100 = 27,27 %
So, the value of the slope of the roof slope, expressed as a percentage, is obtained.
But what if you need to switch from degrees to percentages or vice versa?
You can remember this ratio. 100% is an angle of 45 degrees when the legs of a right triangle are equal to each other, that is, in our case, the height of the slope is equal to the length of its horizontal projection.
In this case, 45° / 100 = 0.45° = 27´ . One percent slope is equal to 27 minutes of arc.
If you approach from the other side, then 100 / 45° = 2.22%. That is, we find that one degree is 2.22% of the slope.
To easily convert values from one to another, you can use the table:
| Value in degrees | Value in % | Value in degrees | Value in % | Value in degrees | Value in % |
| 1° | 2,22% | 16° | 35,55% | 31° | 68,88% |
| 2° | 4,44% | 17° | 37,77% | 32° | 71,11% |
| 3° | 6,66% | 18° | 40,00% | 33° | 73,33% |
| 4° | 8,88% | 19° | 42,22% | 34° | 75,55% |
| 5° | 11,11% | 20° | 44,44% | 35° | 77,77% |
| 6° | 13,33% | 21° | 46,66% | 36° | 80,00% |
| 7° | 15,55% | 22° | 48,88% | 37° | 82,22% |
| 8° | 17,77% | 23° | 51,11% | 38° | 84,44% |
| 9° | 20,00% | 24° | 53,33% | 39° | 86,66% |
| 10° | 22,22% | 25° | 55,55% | 40° | 88,88% |
| 11° | 24,44% | 26° | 57,77% | 41° | 91,11% |
| 12° | 26,66% | 27° | 60,00% | 42° | 93,33% |
| 13° | 28,88% | 28° | 62,22% | 43° | 95,55% |
| 14° | 31,11% | 29° | 64,44% | 44° | 97,77% |
| 15° | 33,33% | 30° | 66,66% | 45° | 100,00% |
For clarity, it will be useful to provide a graphical diagram that very clearly shows the relationship of all the mentioned linear parameters with the angle of the slope and its measurement values.
Diagram A. Interdependence of units of measurement of roof slope angle and permissible types of roofing
We will have to return to this figure when we consider the types of roofing coverings.
It will be even easier to calculate the steepness and angle of inclination of the slope. if you use the built-in calculator below:
Calculator for calculating the steepness of the slope based on the known value of the ridge height
Go to calculations
Dependence of the type of roofing on the steepness of the slope
When planning to build their own house, the owner of the site is probably already assessing both in his head and with his family members what their future home will look like. The roof in this matter, of course, occupies one of the paramount importance. And here it is necessary to take into account the fact that not every roofing material can be used on roof slopes of different steepnesses. To avoid misunderstandings later, it is necessary to foresee this relationship in advance.
Diagram of distribution of roofs by slope steepness
Roofs based on the angle of inclination of the slope can be conditionally divided into flat (slope up to 5°), with a small slope (from 6 to 30°) and steeply inclined, respectively, with a slope angle of more than 30°.
Each type of roof has its own advantages and disadvantages. For example, flat roofs have a minimal area but will require special waterproofing measures. Snow masses do not linger on steep roofs, but they are more susceptible to wind loads due to their “windage”. Likewise, roofing material, due to its own technological or operational characteristics, has certain limitations for use with different slopes.
Let us turn to the figure already discussed earlier ( diagram A ). Black circles with arc-shaped arrows and blue numbers indicate areas of application of various roofing coverings (the arrowhead indicates the minimum permissible value of the slope steepness):
1 – these are shingles, wood chips, natural shingles. The use of reed roofs, still used in the southern regions, also lies in this area.
2 – natural piece tile covering, bitumen-polymer tiles, slate tiles.
3 – rolled materials on a bitumen basis, at least four layers, with an external gravel topping, recessed into a layer of molten mastic.
4 – similar to point 3, but for the reliability of the roof, three layers of rolled material are sufficient.
5 – roll materials similar to those described above (at least three layers), but without external protective gravel topping.
6 – rolled roofing materials glued to hot mastic in at least two layers. Metal tiles, corrugated sheets.
7 – corrugated asbestos-cement sheets (slate) of a unified profile.
8 – tiled clay covering
9 – asbestos-cement sheets with reinforced profile.
10 – roofing sheet steel with flared joints.
11 – slate covering of regular profile.
Thus, if there is a desire to cover the roof with a certain type of roofing material, the slope angle should be planned within the specified limits.
Dependence of the height of the ridge on the angle of inclination of the roof
For those readers who remember their high school trigonometry course well, this section may not seem interesting. They can skip it right away and move on. But those who have forgotten this need to refresh their knowledge about the interdependence of angles and sides in a right triangle.
What is this for? In the case under consideration, the construction of the roof is always based on a right triangle. Its two legs are the length of the projection of the slope onto the horizontal plane (length of the span, half span, etc. - depending on the type of roof) and the height of the slope at the highest point (at the ridge or when moving to the upper rafters - when calculating the lower rafters of the attic roofs). It is clear that there is only one constant value here - the span length. But the height can be changed by varying the angle of the roof.
The table shows two main dependencies, expressed through the tangent and sine of the slope angle. There are other dependencies (via cosine or cotangent), but in this case these two trigonometric functions are enough for us.
| Graphic diagram | Basic trigonometric relations | |
| H - ridge height | ||
| S - roof slope length | ||
| L - half the span length (with a symmetrical gable roof) or span length (with a shed roof) | ||
| α - roof slope angle | ||
| tan α = H/L | Н = L × tan α | |
| sinα = H/S | S = H / sin α | |
Knowing these trigonometric identities, you can solve almost all problems in the preliminary design of a truss structure.
For clarity, a triangle attached to the roof of a house
So, if you need to “dance” from a clearly established height of the skate, then using the ratio tg α = H / L it will be easy to determine the angle.
Using the number obtained by division in the table of tangents, the angle in degrees is found. Trigonometric functions are often included in engineering calculators; they are required in Excel tables (for those who know how to work with this convenient application. However, the calculations there are carried out not in degrees, but in radians). But so that our reader does not have to be distracted by searching for the necessary tables, we present the values of tangents in the range from 1 to 80°.
| Corner | Tangent value | Corner | Tangent value | Corner | Tangent value | Corner | Tangent value |
| tg(1°) | 0.01746 | tg(21°) | 0.38386 | tg(41°) | 0.86929 | tg(61°) | 1.80405 |
| tg(2°) | 0.03492 | tg(22°) | 0.40403 | tg(42°) | 0.9004 | tg(62°) | 1.88073 |
| tg(3°) | 0.05241 | tg(23°) | 0.42447 | tg(43°) | 0.93252 | tg(63°) | 1.96261 |
| tg(4°) | 0.06993 | tg(24°) | 0.44523 | tg(44°) | 0.96569 | tg(64°) | 2.0503 |
| tg(5°) | 0.08749 | tg(25°) | 0.46631 | tg(45°) | 1 | tg(65°) | 2.14451 |
| tg(6°) | 0.1051 | tg(26°) | 0.48773 | tg(46°) | 1.03553 | tg(66°) | 2.24604 |
| tg(7°) | 0.12278 | tg(27°) | 0.50953 | tg(47°) | 1.07237 | tg(67°) | 2.35585 |
| tg(8°) | 0.14054 | tg(28°) | 0.53171 | tg(48°) | 1.11061 | tg(68°) | 2.47509 |
| tg(9°) | 0.15838 | tg(29°) | 0.55431 | tg(49°) | 1.15037 | tg(69°) | 2.60509 |
| tg(10°) | 0.17633 | tg(30°) | 0.57735 | tg(50°) | 1.19175 | tg(70°) | 2.74748 |
| tg(11°) | 0.19438 | tg(31°) | 0.60086 | tg(51°) | 1.2349 | tg(71°) | 2.90421 |
| tan(12°) | 0.21256 | tg(32°) | 0.62487 | tg(52°) | 1.27994 | tg(72°) | 3.07768 |
| tg(13°) | 0.23087 | tg(33°) | 0.64941 | tg(53°) | 1.32704 | tg(73°) | 3.27085 |
| tg(14°) | 0.24933 | tg(34°) | 0.67451 | tg(54°) | 1.37638 | tg(74°) | 3.48741 |
| tg(15°) | 0.26795 | tg(35°) | 0.70021 | tg(55°) | 1.42815 | tg(75°) | 3.73205 |
| tg(16°) | 0.28675 | tg(36°) | 0.72654 | tg(56°) | 1.48256 | tg(76°) | 4.01078 |
| tg(17°) | 0.30573 | tg(37°) | 0.75355 | tg(57°) | 1.53986 | tg(77°) | 4.33148 |
| tg(18°) | 0.32492 | tg(38°) | 0.78129 | tg(58°) | 1.60033 | tg(78°) | 4.70463 |
| tg(19°) | 0.34433 | tg(39°) | 0.80978 | tg(59°) | 1.66428 | tg(79°) | 5.14455 |
| tan(20°) | 0.36397 | tg(40°) | 0.8391 | tg(60°) | 1.73205 | tg(80°) | 5.67128 |
In the case, on the contrary, when the angle of inclination of the roof is taken as a basis, the height of the ridge is determined by the inverse formula:
H = L × tan α
Now, having the values of two legs and the angle of inclination of the roof, it is very easy to calculate the required length of the rafters from the ridge to the eaves overhang. You can apply the Pythagorean theorem
S = √ ( L² + H² )
Or, which is probably easier, since the magnitude of the angle is already known, apply the trigonometric relationship:
S = H / sin α
The value of the sines of angles is in the table below.
| Corner | Sine value | Corner | Sine value | Corner | Sine value | Corner | Sine value |
| sin(1°) | 0.017452 | sin(21°) | 0.358368 | sin(41°) | 0.656059 | sin(61°) | 0.87462 |
| sin(2°) | 0.034899 | sin(22°) | 0.374607 | sin(42°) | 0.669131 | sin(62°) | 0.882948 |
| sin(3°) | 0.052336 | sin(23°) | 0.390731 | sin(43°) | 0.681998 | sin(63°) | 0.891007 |
| sin(4°) | 0.069756 | sin(24°) | 0.406737 | sin(44°) | 0.694658 | sin(64°) | 0.898794 |
| sin(5°) | 0.087156 | sin(25°) | 0.422618 | sin(45°) | 0.707107 | sin(65°) | 0.906308 |
| sin(6°) | 0.104528 | sin(26°) | 0.438371 | sin(46°) | 0.71934 | sin(66°) | 0.913545 |
| sin(7°) | 0.121869 | sin(27°) | 0.45399 | sin(47°) | 0.731354 | sin(67°) | 0.920505 |
| sin(8°) | 0.139173 | sin(28°) | 0.469472 | sin(48°) | 0.743145 | sin(68°) | 0.927184 |
| sin(9°) | 0.156434 | sin(29°) | 0.48481 | sin(49°) | 0.75471 | sin(69°) | 0.93358 |
| sin(10°) | 0.173648 | sin(30°) | 0.5 | sin(50°) | 0.766044 | sin(70°) | 0.939693 |
| sin(11°) | 0.190809 | sin(31°) | 0.515038 | sin(51°) | 0.777146 | sin(71°) | 0.945519 |
| sin(12°) | 0.207912 | sin(32°) | 0.529919 | sin(52°) | 0.788011 | sin(72°) | 0.951057 |
| sin(13°) | 0.224951 | sin(33°) | 0.544639 | sin(53°) | 0.798636 | sin(73°) | 0.956305 |
| sin(14°) | 0.241922 | sin(34°) | 0.559193 | sin(54°) | 0.809017 | sin(74°) | 0.961262 |
| sin(15°) | 0.258819 | sin(35°) | 0.573576 | sin(55°) | 0.819152 | sin(75°) | 0.965926 |
| sin(16°) | 0.275637 | sin(36°) | 0.587785 | sin(56°) | 0.829038 | sin(76°) | 0.970296 |
| sin(17°) | 0.292372 | sin(37°) | 0.601815 | sin(57°) | 0.838671 | sin(77°) | 0.97437 |
| sin(18°) | 0.309017 | sin(38°) | 0.615661 | sin(58°) | 0.848048 | sin(78°) | 0.978148 |
| sin(19°) | 0.325568 | sin(39°) | 0.62932 | sin(59°) | 0.857167 | sin(79°) | 0.981627 |
| sin(20°) | 0.34202 | sin(40°) | 0.642788 | sin(60°) | 0.866025 | sin(80°) | 0.984808 |
For those readers who simply do not want to dive into independent trigonometric calculations, we recommend a built-in calculator that will quickly and accurately determine the length of the roof slope (without taking into account the eaves overhang) based on the available values of the ridge height and the length of the horizontal projection of the slope.
Calculator for calculating the length of the roof slope based on the known value of the ridge height
Skillful use of trigonometric formulas allows, with normal spatial imagination and the ability to make simple drawings, to carry out calculations for more complex roof structures.
Based on the basic ratios, it is easy to divide into triangles and calculate the hip roof
For example, even a hip or mansard roof that seems so “sophisticated” can be divided into a set of triangles, and then all the necessary dimensions can be consistently calculated.
Dependence of the dimensions of the attic room on the angle of inclination of the roof slopes
If the owners of the future house plan to use the attic as a functional room, in other words, to make an attic, then determining the angle of the roof slope acquires quite practical significance.
The greater the slope, the more spacious the attic
There is no need to explain much here - the above diagram clearly shows that the smaller the angle of inclination, the tighter the free space in the attic.
To make it a little clearer, it is better to carry out a similar scheme on a certain scale. Here, for example, is what an attic room will look like in a house with a pediment width of 10 meters. It should be borne in mind that the ceiling height cannot be lower than 2 meters. (Frankly speaking, two meters is not enough for a living space - the ceiling will inevitably “press” on a person. Usually they start from a height of at least 2.5 meters).
For example, a scaled diagram of the attic
You can give the already calculated average values of the room obtained in the attic, depending on the angle of inclination of a conventional gable roof. In addition, the table shows the lengths of the rafters and the area of the roofing material, taking into account 0.5 meters of the roof eaves.
| Roof angle | Ridge height | Ramp length | Useful area of the attic space per 1 meter of building length (with a ceiling height of 2 m) | Roofing area per 1 meter of building length |
| 20 | 1.82 | 5.32 | No | 11.64 |
| 25 | 2.33 | 5.52 | 0.92 | 12.03 |
| 30 | 2.89 | 5.77 | 2.61 | 12.55 |
| 35 | 3.50 | 6.10 | 3.80 | 13.21 |
| 40 | 4.20 | 6.53 | 4.75 | 14.05 |
| 45 | 5.00 | 7.07 | 5.52 | 15.14 |
| 50 | 5.96 | 7.78 | 6.16 | 16.56 |
So, the steeper the slope of the slopes, the more spacious the room. However, this is immediately reflected by a sharp increase in the height of the truss structure, an increase in size, and therefore the mass of parts for its installation. Much more roofing material will be required - the coverage area is also growing rapidly. Plus, we must not forget about the increasing “windage” effect - greater exposure to wind loads. The last chapter of this publication will be devoted to types of external loads.
For comparison, a mansard-type roof provides a gain in usable space even at a lower height
In order to neutralize such negative consequences to a certain extent, designers and builders often use a special mansard roof design - it has already been mentioned in this article. It is more difficult to calculate and manufacture, but it provides a significant gain in the resulting usable area of the attic space with a decrease in the overall height of the building.
Dependence of the magnitude of external loads on the angle of inclination of the roof
Another important application of the calculated roof pitch angle is determining the degree of its influence on the level of external loads falling on the roof structure.
There is an interesting relationship here. You can calculate all the parameters in advance - angles and linear dimensions, but in the end you always end up with detailing. That is, it is necessary to determine from what material the parts and assemblies of the rafter system will be made, what their cross-sectional area should be, the location pitch, the maximum length between adjacent support points, methods of fastening the elements to each other and to the load-bearing walls of the building, and much more.
This is where the loads experienced by the roof structure come to the fore. In addition to your own weight, external influences are of great importance. If we do not take into account seismic loads that are unusual for our region, then we should mainly focus on snow and wind. The size of both is directly related to the angle of the roof to the horizon.
Snow load
It is clear that over the vast territory of the Russian Federation, the average amount of precipitation falling in the form of snow varies significantly by region. Based on the results of many years of observations and calculations, a map of the country’s territory was compiled, which shows eight different zones according to the level of snow load.
Map of distribution of zones in the Russian Federation by snow load
The eighth and last zone is some sparsely populated areas of the Far East, and it need not be particularly considered. The values for other zones are indicated in the table
| Zonal distribution of the territory of the Russian Federation by average snow load | Value in kPa | Value in kg/m² |
| I | 0.8 kPa | 80 kg/m² |
| II | 1.2 kPa | 120 kg/m² |
| III | 1.8 kPa | 180 kg/m² |
| IV | 2.4 kPa | 240 kg/m² |
| V | 3.2 kPa | 320 kg/m² |
| VI | 4.0 kPa | 400 kg/m² |
| VII | 4.8 kPa | 480 kg/m² |
Now, to calculate the specific load for the planned building, you need to use the formula:
Рсн = Рсн.т × μ
RSN.t – the value that we found using the map and table;
Μ – correction factor, which depends on the slope angle α
- at α from 0 to 25° - μ=1
- at α more than 25 and up to 60° - μ=0.7
- when α is more than 60° , the snow load is not taken into account, since the snow should not be kept on the plane of the roof slopes.
For example, a house is being built in Bashkiria. The planned slope of its roof is 35°.
We find from the table - zone V, table value - Рсн.т = 3.2 kPa
We find the final value Рсн = 3.2 × 0.7 = 2.24 kPa
(if the value is needed in kilograms per square meter, then the ratio is used
1 kPa ≈ 100 kg/m²
In our case, it turns out to be 224 kg/m².
Wind load
With wind load, things are much more complicated. The fact is that it can be multidirectional - the wind is able to exert pressure on the roof, pressing it to the base, but at the same time aerodynamic “lifting” forces arise, tending to tear the roof away from the walls.
In addition, wind load affects different parts of the roof unevenly, so knowing only the average level of wind load is not enough. The prevailing wind directions in a given area (“wind rose”), the degree of saturation of the area with obstacles to the spread of wind, the height of the building and surrounding structures, and other criteria are taken into account.
An approximate procedure for calculating wind load is as follows.
First of all, by analogy with previously carried out calculations, the region of the Russian Federation and the corresponding zone are determined on the map.
Distribution of zones in the Russian Federation by wind pressure level
Next, using the table, you can determine the average value of wind pressure Pwt
| Regional distribution of the territory of the Russian Federation by level of average wind load | Ia | I | II | III | IV | V | VI | VII |
| Table value of wind pressure, kg/m² (Рв) | 24 | 32 | 42 | 53 | 67 | 84 | 100 | 120 |
Next, the calculation is carried out using the following formula:
Рв = Рвт × k × c
Pwt – tabular value of wind pressure
k is a coefficient that takes into account the height of the building and the nature of the area around it. It is determined from the table:
| Height of the building (structure) being constructed (z) | Zone A | Zone B | Zone B |
| no more than 5 m | 0.75 | 0.5 | 0.4 |
| from 5 to 10 m | 1.0 | 0.65 | 0.4 |
| from 10 to 20 m | 1.25 | 0.85 | 0.55 |
| from 20 to 40 m | 1.5 | 1.1 | 0.8 |
The table shows three different zones:
- Zone “A” is an open “bare” area, for example, steppe, desert, tundra or forest-tundra, completely exposed to the wind influence of the coast of seas and oceans, large lakes, rivers, and reservoirs.
- Zone “B” - territories of residential villages, small towns, wooded and rough areas, with obstacles to the wind, natural or artificial, about 10 meters high.
- Zone “B” is the territory of large cities with dense buildings, with an average building height of 25 meters and above.
A house is considered to correspond to this particular zone if the specified characteristic features are located within a radius of no less than the height of the building h, multiplied by 30 (for example, for a house of 12 m, the radius of the zone must be at least 360 m). For building heights above 60 m, a circle with a radius of 2000 m is assumed.
c - and this is the same coefficient that depends on the direction of the wind on the building and on the angle of inclination of the roof.
As already mentioned, depending on the direction of the impact and the characteristics of the roof, the wind can produce multidirectional load vectors. The diagram below shows the wind impact zones into which the roof area is usually divided.
Distribution of the roof of a building into zones when calculating wind load
Please note that an intermediate auxiliary quantity e appears. It is taken equal to either 2 × h or b , depending on the wind direction. In any case, the smaller of the two values is taken.
The coefficient c for each zone is taken from the tables, which takes into account the roof slope angle. If for one section both positive and negative coefficient values are provided, then both calculations are carried out, and then the data are summed.
Table of coefficient “ c” for wind directed into the roof slope
| Roof slope angle (α) | F | G | H | I | J |
| 15 ° | — 0,9 | -0.8 | — 0.3 | -0.4 | -1.0 |
| 0.2 | 0.2 | 0.2 | |||
| 30 ° | -0.5 | -0.5 | -0.2 | -0.4 | -0.5 |
| 0.7 | 0.7 | 0.4 | |||
| 45 ° | 0.7 | 0.7 | 0.6 | -0.2 | -0.3 |
| 60 ° | 0.7 | 0.7 | 0.7 | -0.2 | -0.3 |
| 75 ° | 0.8 | 0.8 | 0.8 | -0.2 | -0.3 |
Table of coefficient “ c” for wind directed to the gable part
| Roof slope angle (α) | F | G | H | I |
| 0 ° | -1.8 | -1.3 | -0.7 | -0.5 |
| 15 ° | -1.3 | -1.3 | -0.6 | -0.5 |
| 30 ° | -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, by calculating the wind load, it will be possible to determine the total external force impact for each section of the roof.
Rsum = Rsn + Rv
The resulting value becomes the initial value for determining the parameters of the rafter system. In particular, in the table below, you can find the values of the permissible free length of the rafters between the support points, depending on the cross-section of the beam, the distance between the rafters, the type of material (softwood) and, accordingly, the level of total wind and snow load.
| Wood type | Rafter section (mm) | Distance between adjacent rafters (mm) | |||||
| 300 | 400 | 600 | 300 | 400 | 600 | ||
| total load (snow + wind) | 1.0 kPa | 1.5 kPa | |||||
| Premium quality wood | 40×89 | 3.22 | 2.92 | 2.55 | 2.81 | 2.55 | 2.23 |
| 40×140 | 5.06 | 4.60 | 4.02 | 4.42 | 4.02 | 3.54 | |
| 50×184 | 6.65 | 6.05 | 5.28 | 5.81 | 5.28 | 4.61 | |
| 50×235 | 8.50 | 7.72 | 6.74 | 7.42 | 6.74 | 5.89 | |
| 50×286 | 10.34 | 9.40 | 8.21 | 9.03 | 8.21 | 7.17 | |
| I or II grade | 40×89 | 3.11 | 2.83 | 2.47 | 2.72 | 2.47 | 2.16 |
| 40×140 | 4.90 | 4.45 | 3.89 | 4.28 | 3.89 | 3.40 | |
| 50×184 | 6.44 | 5.85 | 5.11 | 5.62 | 5.11 | 4.41 | |
| 50×235 | 8.22 | 7.47 | 6.50 | 7.18 | 6.52 | 5.39 | |
| 50×286 | 10.00 | 9.06 | 7.40 | 8.74 | 7.66 | 6.25 | |
| III grade | 40×89 | 3.06 | 2.78 | 2.31 | 2.67 | 2.39 | 1.95 |
| 40×140 | 4.67 | 4.04 | 3.30 | 3.95 | 3.42 | 2.79 | |
| 50×184 | 5.68 | 4.92 | 4.02 | 4.80 | 4.16 | 3.40 | |
| 50×235 | 6.95 | 6.02 | 4.91 | 5.87 | 5.08 | 4.15 | |
| 50×286 | 8.06 | 6.98 | 6.70 | 6.81 | 5.90 | 4.82 | |
| total load (snow + wind) | 2.0 kPa | 2.5 kPa | |||||
| Premium quality wood | 40×89 | 4.02 | 3.65 | 3.19 | 3.73 | 3.39 | 2.96 |
| 40×140 | 5.28 | 4.80 | 4.19 | 4.90 | 4.45 | 3.89 | |
| 50×184 | 6.74 | 6.13 | 5.35 | 6.26 | 5.69 | 4.97 | |
| 50×235 | 8.21 | 7.46 | 6.52 | 7.62 | 6.92 | 5.90 | |
| 50×286 | 2.47 | 2.24 | 1.96 | 2.29 | 2.08 | 1.82 | |
| I or II grade | 40×89 | 3.89 | 3.53 | 3.08 | 3.61 | 3.28 | 2.86 |
| 40×140 | 5.11 | 4.64 | 3.89 | 4.74 | 4.31 | 3.52 | |
| 50×184 | 6.52 | 5.82 | 4.75 | 6.06 | 5.27 | 4.30 | |
| 50×235 | 7.80 | 6.76 | 5.52 | 7.06 | 6.11 | 4.99 | |
| 50×286 | 2.43 | 2.11 | 1.72 | 2.21 | 1.91 | 1.56 | |
| III grade | 40×89 | 3.48 | 3.01 | 2.46 | 3.15 | 2.73 | 2.23 |
| 40×140 | 4.23 | 3.67 | 2.99 | 3.83 | 3.32 | 2.71 | |
| 50×184 | 5.18 | 4.48 | 3.66 | 4.68 | 4.06 | 3.31 | |
| 50×235 | 6.01 | 5.20 | 4.25 | 5.43 | 4.71 | 3.84 | |
| 50×286 | 6.52 | 5.82 | 4.75 | 6.06 | 5.27 | 4.30 | |
It is clear that when calculating the cross-section of the rafters, the pitch of their installation and the span length (the distance between the support points), indicators of the total external pressure are taken for the most loaded sections of the roof. table , these are G and N.
To simplify the task of calculating the total load for a site visitor, below is a calculator that will calculate this parameter specifically for the most loaded areas.
Calculator for calculating the total, snow and wind load to determine the required rafter section
Go to calculations
So, it is difficult to underestimate the importance of correct calculation of the angle of inclination of the roof, the influence of this parameter on a number of the most important characteristics of the rafter system, and the entire building as a whole. Although carrying out real architectural calculations, of course, is largely the prerogative of specialists, the ability to navigate basic concepts and carry out simple basic calculations will be very useful for every competent home owner.
And at the end of the article - a video lesson on calculating the rafter system of a conventional gable roof:
Video: calculation and installation of a gable rafter system
Option for calculating roof parameters using a calculator
Step 1. First of all, a website opens in the browser where there is an online calculator. In this case, you can specify a lot of parameters for the future roof. To begin with, select the shape of the roof - for example, pitched.
Choosing the type of roof
Step 2. Next, the site suggests choosing the material with which the finished roof will be covered (metal tiles, slate, etc.).
Selection of roofing material
Step 3. Select the length and width of the roof base, focusing on the image at the bottom of the page - it shows the definitions of the symbols used.
Selecting Basic Values
Step 4. You can immediately specify other values - the parameters of the rafter system, right down to indicating the materials used for its construction. The value of the sheathing pitch and the calculation of the snow load are also selected.
Rafter system parameters
Step 5. The snow load is determined by the region where the structure will be located. There is a convenient diagram map for this.
Selecting a region by snow load
Step 6. For the calculations to be made, click the “Calculate” button.
Click the "Calculate" button
Step 7. As a result, a detailed table will appear on the page indicating the main parameters of the roof, including its angle of inclination.
Calculation results
We also give an approximate calculation of the roof angle depending on the known value of the ridge height. To make calculations, you should measure the width of the pediment (for example, this figure will be 6 m). Next, this value is divided by 2 - the result is 3 m. The height of the ridge in this case should be 1.8 m.
Now you just need to use the formulas known from geometry lessons and find out the tangent of the angle: tgA = a:b = 3:1.8 = 1.67. The angle value based on the tangent value can be found in the Bradis table. In this case, the slope angle will be 58-59 degrees. It can be rounded to 60.
