Calculate the angle between the leg and the hypotenuse. We find the side of the triangle, if the other two are known in three ways, the formulas
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 beforehand, even 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 the angles of inclination, the area of \u200b\u200bthe slopes, etc. How to calculate the angle of the roof correctly? It is from this value that the rest of the parameters of this design will largely depend.
The design and construction of any roof is always a very important and responsible business. Especially when it comes to the roof of a residential building or a roof with a complex shape. But even the usual shed, installed on a nondescript shed or garage, just needs preliminary calculations.
If you do not determine in advance the angle of inclination of the roof, do not find out what optimal height the ridge should have, then there is a great risk of building a roof that will collapse after the first snowfall, or all the finishing coating from it will be torn off even by a moderate wind.
Also, the angle of inclination 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 finish.
Prices for various types of roof ridges
Roofing ridge
Units
Remembering the geometry that everyone learned 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 also find another option - the angle is indicated as a percentage (here we mean the aspect ratio).
Generally, slope angle is the angle formed by two intersecting planes- overlapping and directly the slope of the roof. It can only be sharp, that is, lie in the range of 0-90 degrees.
On a note! Very steep slopes, the angle of which is more than 50 degrees, are extremely rare in their pure form. Usually they are used only for the decoration of roofs, they may be present in attics.
As for measuring the angles of the roof in degrees, then everything is simple - everyone who studied geometry at school has this knowledge. It is enough to sketch a roof diagram on paper and use a protractor to determine the angle.
As for the percentages, then 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%. Thus, the percentage can be calculated.
On a note! At a percentage of 1, a 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 minutes of arc.
Table of values - degrees, minutes, percent
What factors affect 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 to 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. At some point, they may play their part. Determine the appropriate angle of inclination of the roof should be, knowing:
- types of materials from which the roof pie will be built, starting from the truss system and ending with the exterior finish;
- climate conditions in the area (wind load, prevailing wind direction, precipitation, etc.);
- the shape of the future building, its height, design;
- purpose of the building, options for using the attic space.
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, with a strong wind, the roof is more likely to resist and not be 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 shed 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 in general, 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 larger the roof slope parameters, the more materials will be required to create it. The cost increases by at least 20%.
Pitch angles and roofing materials
Not only climatic conditions will have a significant impact on the shape and angle of the slopes. An important role is played by the materials used for construction, in particular - roofing.
Table. Optimum slope angles for roofs of various materials.
On a note! The lower the roof slope, the smaller the pitch used to create the crate.
Prices for metal tiles
metal tile
The height of the skate also depends on the angle of the slope.
When calculating any roof, a rectangular triangle is always taken as a guideline, where the legs are the height of the slope at the top point, that is, at the ridge or the transition from the lower part of the entire rafter system to the top (in the case of mansard 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.
Knowing the formulas from trigonometry will help to design the roof: tgA \u003d H / L, sinA \u003d H / S, H \u003d LхtgA, S \u003d 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 shed 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 by the first formula. You can find the angle using the table of tangents. If the calculation is based on the angle of the roof, then you can find the ridge height parameter 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.
A right triangle is found in reality on almost every corner. Knowledge of the properties of this figure, as well as the ability to calculate its area, will undoubtedly be useful to you not only for solving problems in geometry, but also in life situations.
triangle geometry
In elementary geometry, a right triangle is a figure that consists of three connected segments that form three angles (two acute and one straight). A right triangle is an original figure, characterized by a number of important properties that form the foundation of trigonometry. Unlike an ordinary triangle, the sides of a rectangular figure have their own names:
- The hypotenuse is the longest side of a triangle that lies opposite the right angle.
- Legs - segments that form a right angle. Depending on the angle under consideration, the leg may be adjacent to it (forming this angle with the hypotenuse) or opposite (lying opposite the angle). There are no legs for non-rectangular triangles.
It is the ratio of the legs and hypotenuse that forms the basis of trigonometry: sines, tangents and secants are defined as the ratio of the sides of a right triangle.
Right triangle in reality
This figure is widely used in reality. Triangles are used in design and technology, so the calculation of the area of \u200b\u200bthe figure has to be done by engineers, architects and designers. The bases of tetrahedra or prisms have the shape of a triangle - three-dimensional figures that are easy to meet in everyday life. In addition, a square is the simplest representation of a "flat" right triangle in reality. A square is a locksmith, drawing, construction and carpentry tool that is used to build corners by both schoolchildren and engineers.
Area of a triangle
The area of a geometric figure is a quantitative estimate of how much of the plane is bounded by the sides of a triangle. The area of an ordinary triangle can be found in five ways, using Heron's formula or operating in calculations with such variables as the base, side, angle and radius of the inscribed or circumscribed circle. The simplest area formula is expressed as:
where a is the side of the triangle, h is its height.
The formula for calculating the area of a right triangle is even simpler:
where a and b are legs.
Working with our online calculator, you can calculate the area of a triangle using three pairs of parameters:
- two legs;
- leg and adjacent angle;
- leg and opposite angle.
In tasks or everyday situations, you will be given different combinations of variables, so this form of calculator allows you to calculate the area of a triangle in several ways. Let's look at a couple of examples.
Real life examples
Ceramic tile
Let's say you want to line the walls of the kitchen with ceramic tiles, which have the shape of a right triangle. In order to determine the consumption of tiles, you must find out the area of \u200b\u200bone element of the cladding and the total area of \u200b\u200bthe surface to be treated. Suppose you need to process 7 square meters. The length of the legs of one element is 19 cm each, then the area of \u200b\u200bthe tile will be equal to:
This means that the area of one element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that to finish 7 square meters of a wall you will need 7 / 0.01805 = 387 facing tiles.
school task
Suppose that in a school geometry problem it is required to find the area of a right triangle, knowing only that the side of one leg is 5 cm, and the value of the opposite angle is 30 degrees. Our online calculator is accompanied by an illustration showing the sides and angles of a right triangle. If side a = 5 cm, then its opposite angle is the angle alpha, equal to 30 degrees. Enter this data into the calculator form and get the result:
Thus, the calculator not only calculates the area of a given triangle, but also determines the length of the adjacent leg and hypotenuse, as well as the value of the second angle.
Conclusion
Rectangular triangles are found in our lives literally on every corner. Determining the area of such figures will be useful to you not only when solving school assignments in geometry, but also in everyday and professional activities.
ANDREY PROKIP: “MY LOVE IS RUSSIAN ECOLOGY. YOU SHOULD INVEST IN IT!”
On September 4-5, the ecological forum "Climatic shape of cities" was held. The initiator of the organization of the event is the C40 organization, which was founded in 2005 by the UN. The main task of the form and cities is to control climate change in cities.
As practice has shown, unlike social events and "meetings in nightclubs", there were few deputies and public personalities. Among those who really revealed concern about the environmental situation was Prokip Adrey Zinovievich. He took an active part in all plenary sessions together with Ruslan Edelgeriev, Special Representative of the President of the Russian Federation for Climate Issues, Petr Biryukov, Deputy Mayor of Moscow for Housing and Communal Services, as well as foreign representatives - the Mayor of the Italian city of Savona - Ilario Caprioglio. The participants presented their projects and also discussed strategies to keep the rise in global temperatures, as well as proposed practical solutions for sustainable urban development.
ANDREY PROKIP ABOUT SHASHLIKS, DEPUTY AND GREEN CONSTRUCTION
Of particular interest to the Russian side was the speech of the speakers, among whom were European architects, scientists and the mayor of Savona. The topic of the speech was the TOP direction - "green construction". As Andrei Prokip himself stated, “it is important to correctly redistribute resources, as well as take into account the standards of European construction for such a metropolis as Moscow. It is necessary that Russia at the federal level take a course towards “green financing”, especially since it is economically feasible and, as practice shows, profitable.” He also expressed concern about the deterioration of the health of Russians in connection with environmental disasters and non-compliance with environmental standards for waste disposal by large and small industrial enterprises. He also confirmed his fears thanks to the speech of Francesco Zambon, WHO European Bureau of Health Investment Professor.
With characteristic humor, Andrey turned to famous people who were invited to the forum, but never showed up, with a call “to remember nature, not only when they want barbecue or go fishing. After all, it is on the benevolence of nature that the health of the whole people depends, which, unfortunately, includes them.
In addition to passionate speeches about Andrey Zinovievich's new "mistress-nature" and the importance of taking responsibility for the environment, the plenary session on the topic "How to educate a new generation" became a significant event of the forum. The forum participants were unanimous in their opinion that it is necessary to educate not only children, but also the adult generation. It is very important to bring up responsibility to nature in everyday behavior, as well as in business.
A special project “Learning to live in a civilized way” will be launched for Moscow. This is an educational project for all segments of the population and age categories. But no matter how wonderful the theory and good intentions are, the saying “until the roasted rooster pecks, the fool will not cross himself” is still relevant for Russia.
According to Timothy Netter, a famous theater director, art can change everything. In one of his speeches, he spoke about how the idea of preserving nature should be presented in theater and cinema, and how important it is to educate people through art to be responsible for what will happen to us and nature tomorrow.
The attention of rentv operators and Andrey Prokirp was attracted by students of Russian universities, who presented a project on environmentally friendly technology for the production of containers that are resistant to moisture and temperature. This is a very urgent problem, as laws are being passed around the world against plastic containers, which, by the way, decompose for more than 30 years, pollute the soil and cause the death of animals.
It is inspiring that Moscow is one of the 94 cities participating in the C40 organization and for the third time the forum has been held, which every year attracts the attention of more and more famous personalities and citizens.
The first are segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the 90 degree angle. A Pythagorean triangle is one whose sides are equal to natural numbers; their lengths in this case are called the "Pythagorean triple".
egyptian triangle
In order for the current generation to learn geometry in the form in which it is taught at school now, it has been developed for several centuries. The fundamental point is the Pythagorean theorem. The sides of a rectangle are known to the whole world) are 3, 4, 5.
Few people are not familiar with the phrase "Pythagorean pants are equal in all directions." However, in fact, the theorem sounds like this: c 2 (the square of the hypotenuse) \u003d a 2 + b 2 (the sum of the squares of the legs).
Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called "Egyptian". It is interesting that which is inscribed in the figure is equal to one. The name arose around the 5th century BC, when Greek philosophers traveled to Egypt.
When building the pyramids, architects and surveyors used the ratio 3:4:5. Such structures turned out to be proportional, pleasant to look at and spacious, and also rarely collapsed.
In order to build a right angle, the builders used a rope on which 12 knots were tied. In this case, the probability of constructing a right-angled triangle increased to 95%.
Signs of equality of figures
- An acute angle in a right triangle and a large side, which are equal to the same elements in the second triangle, is an indisputable sign of the equality of the figures. Taking into account the sum of the angles, it is easy to prove that the second acute angles are also equal. Thus, the triangles are identical in the second criterion.
- When two figures are superimposed on each other, we rotate them in such a way that, when combined, they become one isosceles triangle. According to its property, the sides, or rather, the hypotenuses, are equal, as well as the angles at the base, which means that these figures are the same.
By the first sign, it is very easy to prove that the triangles are really equal, the main thing is that the two smaller sides (i.e., the legs) are equal to each other.
The triangles will be the same according to the II sign, the essence of which is the equality of the leg and the acute angle.
Right angle triangle properties
The height, which was lowered from a right angle, divides the figure into two equal parts.
The sides of a right triangle and its median are easy to recognize by the rule: the median, which is lowered to the hypotenuse, is equal to half of it. can be found both by Heron's formula and by the statement that it is equal to half the product of the legs.
In a right triangle, the properties of angles of 30 o, 45 o and 60 o apply.
- At an angle that is 30 °, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
- If the angle is 45o, then the second acute angle is also 45o. This suggests that the triangle is isosceles, and its legs are the same.
- The property of an angle of 60 degrees is that the third angle has a measure of 30 degrees.
The area is easy to find by one of three formulas:
- through the height and the side on which it descends;
- according to Heron's formula;
- along the sides and the angle between them.
The sides of a right triangle, or rather the legs, converge with two heights. In order to find the third, it is necessary to consider the resulting triangle, and then, using the Pythagorean theorem, calculate the required length. In addition to this formula, there is also the ratio of twice the area and the length of the hypotenuse. The most common expression among students is the first, as it requires less calculations.
Theorems that apply to a right triangle
The geometry of a right triangle includes the use of theorems such as:
A triangle is a geometric number made up of three segments that connect three points that do not lie on the same line. The points that form a triangle are called its points, and the segments are side by side.
Depending on the type of triangle (rectangular, monochrome, etc.) you can calculate the side of the triangle in different ways, depending on the input data and the conditions of the problem.
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To calculate the sides of a right triangle, the Pythagorean theorem is used, according to which the square of the hypotenuse is equal to the sum of the squares of the leg.
If we label the legs with "a" and "b" and the hypotenuse with "c", then pages can be found with the following formulas:
If the acute angles of a right triangle (a and b) are known, its sides can be found with the following formulas:
cropped triangle
A triangle is called an equilateral triangle in which both sides are the same.
How to find the hypotenuse in two legs
If the letter "a" is identical to the same page, "b" is the base, "b" is the corner opposite the base, "a" is the adjacent corner, the following formulas can be used to calculate pages:
Two corners and side
If one page (c) and two angles (a and b) of any triangle are known, the sine formula is used to calculate the remaining pages:
You must find the third value y = 180 - (a + b) because
the sum of all the angles of a triangle is 180°; 
Two sides and an angle
If two sides of a triangle (a and b) and the angle between them (y) are known, the cosine theorem can be used to calculate the third side.
How to determine the perimeter of a right triangle
A triangular triangle is a triangle, one of which is 90 degrees, and the other two are acute. payment perimeter such triangle depending on the amount of known information about it.
You will need it
- Depending on the occasion, skills 2 of the three sides of the triangle, as well as one of its sharp corners.
instructions
first Method 1. If all three pages are known triangle. Then, whether perpendicular or not triangular, the perimeter is calculated as: P = A + B + C, where possible, c is the hypotenuse; a and b are legs.
second Method 2.
If a rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated using the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.
the third Method 3. Let the hypotenuse be c and an acute angle? Given a right triangle, it will be possible to find the perimeter in this way: P = (1 + sin?
fourth Method 4. They say that in the right triangle the length of one leg is equal to a and, on the contrary, has an acute angle. Then calculate perimeter this triangle will be performed according to the formula: P = a * (1 / tg?
1 / son? + 1)
fifth Method 5.
Triangle Online Calculation
Let our leg lead and be included in it, then the range will be calculated as: P = A * (1 / CTG + 1 / + 1 cos?)
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The Pythagorean theorem is the basis of any mathematics. Specifies the relationship between the sides of a true triangle. Now there are 367 proofs of this theorem.
instructions
first The classic school formulation of the Pythagorean theorem sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs.
To find the hypotenuse in a right triangle of two Catets, you must turn to square the length of the legs, assemble them, and take the square root of the sum. In the original formulation of his statement, the market is based on the hypotenuse, equal to the sum of the squares of 2 squares produced by Catete. However, the modern algebraic formulation does not require the introduction of a domain representation.
second For example, a right triangle whose legs are 7 cm and 8 cm.
Then, according to the Pythagorean theorem, the square hypotenuse is R + S = 49 + 64 = 113 cm. The hypotenuse is equal to the square root of 113.
Angles of a right triangle
The result was an unreasonable number.
the third If the triangles are legs 3 and 4, then the hypotenuse = 25 = 5. When you take the square root, you get a natural number. The numbers 3, 4, 5 form a Pygagorean triple, since they satisfy the relation x? +Y? = Z, which is natural.
Other examples of a Pythagorean triplet are: 6, 8, 10; 5, 12, 13; 15, 20, 25; 9, 40, 41.
fourth In this case, if the legs are identical to each other, the Pythagorean theorem turns into a more primitive equation. For example, let such a hand be equal to the number A and the hypotenuse is defined for C, and then c? = Ap + Ap, C = 2A2, C = A? 2. In this case, you don't need A.
fifth The Pythagorean theorem is a special case that is larger than the general cosine theorem, which establishes a relationship between the three sides of a triangle for any angle between two of them.
Tip 2: How to determine the hypotenuse for legs and angles
The hypotenuse is called the side in a right triangle that is opposite the 90 degree angle.
instructions
first In the case of well-known catheters, as well as an acute angle of a right triangle, the hypotenuse size can be equal to the ratio of the leg to the cosine / sine of this angle, if the angle was opposite / e include: H = C1 (or C2) / sin, H = C1 (or С2 ?) / cos ?. Example: Let ABC be given an irregular triangle with hypotenuse AB and right angle C.
Let B be 60 degrees and A 30 degrees. The length of the stem BC is 8 cm. The length of the hypotenuse AB should be found. To do this, you can use one of the above methods: AB = BC / cos60 = 8 cm. AB = BC / sin30 = 8 cm.
The hypotenuse is the longest side of the rectangle triangle. It is located at a right angle. Method for finding the hypotenuse of a rectangle triangle depending on the source data.
instructions
first If your legs are perpendicular triangle, then the length of the hypotenuse of the rectangle triangle can be found by the Pythagorean analogue - the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs: c2 = a2 + b2, where a and b are the length of the legs of the right triangle .
second If it is known and one of the legs is at an acute angle, the formula for finding the hypotenuse will depend on the presence or absence at a certain angle with respect to the known leg - adjacent (the leg is located near), or vice versa (the opposite case is located nego.V of the specified angle is equal to the fraction leg hypotenuse in cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio of sinusoidal angles: da = a / sin.
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Helpful Hints
An angled triangle whose sides are connected as 3:4:5, called the Egyptian delta, due to the fact that these figures were widely used by the architects of ancient Egypt.
This is also the simplest example of Jeron's triangles, with pages and area represented as integers.
A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other side is called the legs.
If you want to find how a right triangle is formed by some properties of regular triangles, namely the fact that the sum of the acute angles is 90°, which is used, and the fact that the length of the opposite leg is half the hypotenuse is 30°.
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cropped triangle
One of the properties of an equal triangle is that its two angles are the same.
To calculate the angle of a right equilateral triangle, you need to know that:
- It's no worse than 90°.
- The values of acute angles are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.
Angles α and β are 45°.
If the known value of one of the acute angles is known, the other can be found using the formula: β = 180º-90º-α or α = 180º-90º-β.
This ratio is most commonly used if one of the angles is 60° or 30°.
Key Concepts
The sum of the interior angles of a triangle is 180°.
Because it's one level, two stay sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The acute angle values of a right triangle can be calculated from the mean - with a line from a point on the opposite side of the triangle, and the height - the line is a perpendicular drawn from the hypotenuse at a right angle.
Let the median extend from the right corner to the middle of the hypotenuse, and h be the height. In this case it turns out that: 
- sinα = b / (2 * s); sin β = a / (2 * s).
- cosα = a / (2 * s); cos β = b / (2 * s).
- sinα = h / b; sin β = h / a.
Two pages
If the lengths of the hypotenuse and one of the legs are known in a right triangle or from two sides, then trigonometric identities are used to determine the values of acute angles:
- α=arcsin(a/c), β=arcsin(b/c).
- α=arcos(b/c), β=arcos(a/c).
- α = arctan (a / b), β = arctan (b / a).
Length of a right triangle
Area and Area of a Triangle
perimeter
The circumference of any triangle is equal to the sum of the lengths of the three sides. The general formula for finding a triangular triangle is:
where P is the circumference of the triangle, a, b and c are its sides.
Perimeter of an equal triangle can be found by successively combining the lengths of its sides, or multiplying the side length by 2 and adding the length of the base to the product.
The general formula for finding an equilibrium triangle will look like this:
where P is the perimeter of an equal triangle, but either b, b are the base.
Perimeter of an equilateral triangle can be found by successively combining the lengths of its sides, or by multiplying the length of any page by 3.
The general formula for finding the rim of equilateral triangles would look like this:
where P is the perimeter of an equilateral triangle, a is any of its sides.
region
If you want to measure the area of a triangle, you can compare it to a parallelogram. Consider triangle ABC:
If we take the same triangle and fix it so that we get a parallelogram, we get a parallelogram with the same height and base as this triangle:
In this case, the common side of the triangles is folded together along the diagonal of the molded parallelogram.
From the properties of a parallelogram. It is known that the diagonals of a parallelogram are always divided into two equal triangles, then the surface of each triangle is equal to half the range of the parallelogram.
Since the area of the parallelogram is the product of its base height, the area of the triangle will be half that product. So for ΔABC the area will be the same
Now consider a right triangle:
Two identical right triangles can be bent into a rectangle if it leans against them, which is every other hypotenuse.
Since the surface of the rectangle coincides with the surface of the adjacent sides, the area of this triangle is the same:
From this we can conclude that the surface of any right triangle is equal to the product of the legs divided by 2.
From these examples, we can conclude that the surface of each triangle is the same as the product of the length, and the height is reduced to the base divided by 2.
The general formula for finding the area of a triangle would look like this:
where S is the area of the triangle, but its base, but the height falls to the bottom a.