A triangle is called a right triangle if one of its angles is 90º. The side opposite the right angle is called the hypotenuse, and the other two are called the legs.

To find the angle in right triangle, some properties of right triangles are used, namely: the fact that the sum of the acute angles is equal to 90º, and also that opposite the leg, the length of which is half the length of the hypotenuse, lies an angle equal to 30º.

Quick navigation through the article

Isosceles triangle

One of the properties of an isosceles triangle is that its two angles are equal. To calculate the angles of a right isosceles triangle you need to know that:

• A right angle is 90º.
• The values ​​of acute angles are determined by the formula: (180º-90º)/2=45º, i.e. angles α and β are equal to 45º.

If the size of one of the acute angles is known, the second can be found using the formula: β=180º-90º-α, or α=180º-90º-β. Most often this ratio is used if one of the angles is 60º or 30º.

Key Concepts

The sum of the interior angles of a triangle is 180º. Since one angle is right, the remaining two will be acute. To find them you need to know that:

Other ways

The values ​​of the acute angles of a right triangle can be calculated by knowing the value of the median - a line drawn from the vertex to the opposite side of the triangle, and the height - a straight line, which is a perpendicular dropped from a right angle to the hypotenuse. Let s be the median drawn from the right angle to the middle of the hypotenuse, h be the height. In this case it turns out that:

• sin α=b/(2*s); sin β =a/(2*s).
• cos α=a/(2*s); cos β=b/(2*s).
• sin α=h/b; sin β =h/a.

Two sides

If the lengths of the hypotenuse and one of the legs, or two sides, are known in a right triangle, trigonometric identities are used to find the values ​​of the acute angles:

• α=arcsin(a/c), β=arcsin(b/c).
• α=arcos(b/c), β=arcos(a/c).
• α=arctg(a/b), β=arctg(b/a).

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.

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.

If you do not determine in advance the angle of inclination of the roof, and 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.

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.

Prices for different types of roofing ridges

Roofing ridge

Units of measurement

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).

Generally, 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.

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.

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.

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 factors influence the angle of inclination?

The angle of inclination of any roof is greatly influenced by large number 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.

In those regions where there is a strong wind load, it is recommended to build a roof with one slope and a slight 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.

Note! The greater the roof slope parameters, the greater the amount of materials required to create it. The cost increases by at least 20%.

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.

Table. Optimal slope angles for roofs made of various materials.

Note! The lower the roof slope, the smaller the pitch used when creating the sheathing.

Prices for metal tiles

Metal tiles

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 specific 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.

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 (for a gable roof) or the entire length (for a single-pitch roof), S – the length of the slope itself. For example, if the exact 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.

Online calculator.
Solving triangles.

Solving a triangle is finding all its six elements (i.e., three sides and three angles) from any three given elements that define the triangle.

This mathematical program finds the side \(c\), angles \(\alpha \) and \(\beta \) from user-specified sides \(a, b\) and the angle between them \(\gamma \)

The program not only gives the answer to the problem, but also displays the process of finding a solution.

This online calculator may be useful for high school students secondary schools in preparation for tests and exams, when testing knowledge before the Unified State Exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get it done as quickly as possible? homework in mathematics or algebra? In this case, you can also use our programs with detailed solutions.

In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.

If you are not familiar with the rules for entering numbers, we recommend that you familiarize yourself with them.

Rules for entering numbers

Numbers can be specified not only as whole numbers, but also as fractions.
Whole and fractional part in decimal fractions can be separated by either a period or a comma.
For example, you can enter decimals so 2.5 or so 2.5

Enter the sides \(a, b\) and the angle between them \(\gamma \) Solve triangle

It was discovered that some scripts necessary to solve this problem were not loaded, and the program may not work.
You may have AdBlock enabled.
In this case, disable it and refresh the page.

JavaScript is disabled in your browser.
For the solution to appear, you need to enable JavaScript.
Here are instructions on how to enable JavaScript in your browser.

Because There are a lot of people willing to solve the problem, your request has been queued.
In a few seconds the solution will appear below.
Please wait sec...

If you noticed an error in the solution, then you can write about this in the Feedback Form.
Don't forget indicate which task you decide what enter in the fields.

Our games, puzzles, emulators:

A little theory.

Theorem of sines

Theorem

The sides of a triangle are proportional to the sines of the opposite angles:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) = \frac(c)(\sin C) $$

Cosine theorem

Theorem
Let AB = c, BC = a, CA = b in triangle ABC. Then
Square side of triangle equal to the sum squares of the other two sides minus twice the product of these sides multiplied by the cosine of the angle between them.
$$ a^2 = b^2+c^2-2ba \cos A $$

Solving triangles

Solving a triangle is finding all its six elements (i.e., three sides and three angles) from any three given elements that define the triangle.

Let's look at three problems involving solving a triangle. In this case, we will use the following notation for the sides of triangle ABC: AB = c, BC = a, CA = b.

Solving a triangle using two sides and the angle between them

Given: \(a, b, \angle C\). Find \(c, \angle A, \angle B\)

Solution
1. Using the cosine theorem, we find \(c\):

$$ c = \sqrt( a^2+b^2-2ab \cos C ) $$ 2. Using the cosine theorem, we have:
$$ \cos A = \frac( b^2+c^2-a^2 )(2bc) $$

3. \(\angle B = 180^\circ -\angle A -\angle C\)

Solving a triangle by side and adjacent angles

Given: \(a, \angle B, \angle C\). Find \(\angle A, b, c\)

Solution
1. \(\angle A = 180^\circ -\angle B -\angle C\)

2. Using the sine theorem, we calculate b and c:
$$ b = a \frac(\sin B)(\sin A), \quad c = a \frac(\sin C)(\sin A) $$

Solving a triangle using three sides

Given: \(a, b, c\). Find \(\angle A, \angle B, \angle C\)

Solution
1. Using the cosine theorem we obtain:
$$ \cos A = \frac(b^2+c^2-a^2)(2bc) $$

Using \(\cos A\) we find \(\angle A\) using a microcalculator or using a table.

2. Similarly, we find angle B.
3. \(\angle C = 180^\circ -\angle A -\angle B\)

Solving a triangle given two sides and an angle opposite a known side

Given: \(a, b, \angle A\). Find \(c, \angle B, \angle C\)

Solution
1. Using the theorem of sines, we find \(\sin B\) we get:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) \Rightarrow \sin B = \frac(b)(a) \cdot \sin A $$

Let's introduce the notation: \(D = \frac(b)(a) \cdot \sin A \). Depending on the number D, the following cases are possible:
If D > 1, such a triangle does not exist, because \(\sin B\) cannot be greater than 1
If D = 1, there is a unique \(\angle B: \quad \sin B = 1 \Rightarrow \angle B = 90^\circ \)
If D If D 2. \(\angle C = 180^\circ -\angle A -\angle B\)

3. Using the sine theorem, we calculate the side c:
$$ c = a \frac(\sin C)(\sin A) $$

Books (textbooks) Abstracts of the Unified State Examination and the Unified State Examination tests online Games, puzzles Plotting graphs of functions Spelling dictionary of the Russian language Dictionary of youth slang Catalog of Russian schools Catalog of secondary educational institutions of Russia Catalog of Russian universities List of tasks

In geometry there are often problems related to the sides of triangles. For example, it is often necessary to find a side of a triangle if the other two are known.

Triangles are isosceles, equilateral and unequal. From all the variety, for the first example we will choose a rectangular one (in such a triangle, one of the angles is 90°, the sides adjacent to it are called legs, and the third is the hypotenuse).

Quick navigation through the article

Length of the sides of a right triangle

The solution to the problem follows from the theorem of the great mathematician Pythagoras. It says that the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse: a²+b²=c²

• Find the square of the leg length a;
• Find the square of leg b;
• We put them together;
• From the obtained result we extract the second root.

Example: a=4, b=3, c=?

• a²=4²=16;
• b² =3²=9;
• 16+9=25;
• √25=5. That is, the length of the hypotenuse of this triangle is 5.

If the triangle does not have a right angle, then the lengths of the two sides are not enough. For this, a third parameter is needed: this can be an angle, the height of the triangle, the radius of the circle inscribed in it, etc.

If the perimeter is known

In this case, the task is even simpler. The perimeter (P) is the sum of all sides of the triangle: P=a+b+c. Thus, by solving a simple mathematical equation we get the result.

Example: P=18, a=7, b=6, c=?

1) We solve the equation by moving all known parameters to one side of the equal sign:

2) Substitute the values ​​instead of them and calculate the third side:

c=18-7-6=5, total: the third side of the triangle is 5.

If the angle is known

To calculate the third side of a triangle given an angle and two other sides, the solution boils down to calculating trigonometric equation. Knowing the relationship between the sides of the triangle and the sine of the angle, it is easy to calculate the third side. To do this, you need to square both sides and add their results together. Then subtract from the resulting product the product of the sides multiplied by the cosine of the angle: C=√(a²+b²-a*b*cosα)

If the area is known

In this case, one formula will not do.

1) First, calculate sin γ, expressing it from the formula for the area of ​​a triangle:

sin γ= 2S/(a*b)

2) By the following formula calculate the cosine of the same angle:

sin² α + cos² α=1

cos α=√(1 — sin² α)=√(1- (2S/(a*b))²)

3) And again we use the theorem of sines:

C=√((a²+b²)-a*b*cosα)

C=√((a²+b²)-a*b*√(1- (S/(a*b))²))

Substituting the values ​​of the variables into this equation, we obtain the answer to the problem.

Enter known triangle data

Side a

Side b

Side c

Angle A in degrees

Angle B in degrees

Angle C in degrees

Median on side a

Median to side b

Median on side c

Height on side a

Height on side b

Height on side c

Coordinates of vertex A

X Y

Vertex B coordinates

X Y

Coordinates of vertex C

X Y

Area of ​​triangle S

Semi-perimeter of the sides of a triangle p

We present to you a calculator that allows you to calculate all possible...

I would like to draw your attention to the fact that This is a universal bot. It calculates all the parameters of an arbitrary triangle, with an arbitrary given parameters. You won't find a bot like this anywhere.

Do you know the side and two heights? or two sides and a median? Or the bisector of two angles and the base of a triangle?

For any requests, we can obtain the correct calculation of the triangle parameters.

You do not need to look for formulas and do the calculations yourself. Everything has already been done for you.

Create a request and get an accurate answer.

An arbitrary triangle is shown. Let’s immediately clarify how and what is indicated, so that in the future there will be no confusion and errors in calculations.

The sides opposite to any angle are also called only with a small letter. That is, opposite angle A lies side of the triangle, side C is opposite angle C.

ma is the medina falling on side a; accordingly, there are also medians mb and mc falling on the corresponding sides.

lb is the bisector falling on side b, respectively, there are also bisectors la and lc falling on the corresponding sides.

hb is the height falling on side b, respectively, there are also heights ha and hc falling on the corresponding sides.

Well, secondly, remember that a triangle is a figure in which there is fundamental rule:

The sum of any(!) two sides must be greaterthird.

So don't be surprised if you get an error P With such data, a triangle does not exist when trying to calculate the parameters of a triangle with sides 3, 3 and 7.

Syntax

For those who allow XMPP clients, the request is this treug<список параметров>

For site users, everything is done on this page.

List of parameters - parameters that are known, separated by semicolons

the parameter is written as parameter=value

For example, if side a with the value 10 is known, then we write a=10

Moreover, the values ​​can be not only in the form of a real number, but also, for example, as the result of some kind of expression

And here is the list of parameters that may appear in the calculations.

Side a

Side b

Side c

Semi-perimeter p

Angle A

Angle B

Angle C

Area of ​​triangle S

Height ha on side a

Height hb on side b

Height hc on side c

Median ma to side a

Median mb to side b

Median mc to side c

Vertex coordinates (xa,ya) (xb,yb) (xc,yc)

Examples

we write treug a=8;C=70;ha=2

Triangle parameters according to given parameters

Side a = 8

Side b = 2.1283555449519

Side c = 7.5420719851515

Semi-perimeter p = 8.8352137650517

Angle A = 2.1882518638666 in degrees 125.37759631119

Angle B = 2.873202966917 in degrees 164.62240368881

Angle C = 1.221730476396 in 70 degrees

Area of ​​the triangle S = 8

Height ha on side a = 2

Height hb on side b = 7.5175409662872

Height hc on side c = 2.1214329472723

Median ma per side a = 3.8348889915443

Median mb per side b = 7.7012304590352

Median mc per side c = 4.4770789813853

That's all, all the parameters of the triangle.

The question is why we named the side A, not V or With? This does not affect the decision. The main thing is to withstand the condition that I have already mentioned" The sides opposite to any angle are called the same, only with a small letter"And then draw a triangle in your mind and apply it to the question asked.

It could be taken instead A V, but then the adjacent angle will not be WITH A A well, the height will be hb. The result if you check will be the same.

For example, like this (xa,ya) =3.4 (xb,yb) =-6.14 (xc,yc)=-6,-3

write a request treug xa=3;ya=4;xb=-6;yb=14;xc=-6;yc=-3

and we get

Triangle parameters according to given parameters

Side a = 17

Side b = 11.401754250991

Side c = 13.453624047073

Semi-perimeter p = 20.927689149032

Angle A = 1.4990243938603 in degrees 85.887771155351

Angle B = 0.73281510178655 in degrees 41.987212495819

Angle C = 0.90975315794426 in degrees 52.125016348905

Area of ​​the triangle S = 76.5

Height ha on side a = 9

Height hb on side b = 13.418987695398

Height hc on side c = 11.372400437582

Median ma per side a = 9.1241437954466

Median mb per side b = 14.230249470757

Median mc per side c = 12.816005617976

Happy calculations!!