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Calculate the angle by knowing the length of the sides. Ways to find an angle in a right-angled triangle - calculation formulas. Online calculator. Solving triangles

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 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 mark the legs with the letters "a" and "b" and the hypotenuse with "c", then the pages can be found with the following formulas:

If the acute angles of a right-angled 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 opposite corner of 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 should find the third value y = 180 - (a + b) because

the sum of all the angles of the triangle is 180 °;

Two sides and an angle

If you know the two sides of the triangle (a and b) and the angle between them (y), 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 sharp. payment perimeter such triangle depending on the amount of known information about it.

You need it

• Depending on the case, skills are 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, regardless, perpendicular or non-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 the rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated by the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.

third Method 3. Let the hypotenuse c and an acute angle? Given a right-angled triangle, it will be possible to detect the perimeter in this way: P = (1 + sin?

fourth Method 4. It is said 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 is triangle will be performed according to the formula: P = a * (1 / tg?

1 / son? + 1)

fifth Method 5.

Online triangle 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 foundation of any mathematics. Defines the relationship between the sides of a true triangle. Now 367 proofs of this theorem are indicated.

instructions

first The classical 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, collect them and take Square root from the amount. In the original formulation of his statement, the market is based on a hypotenuse equal to the sum 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-angled triangle whose legs are 7 cm and 8 cm.

Then, according to the Pythagorean theorem, the square hypotenuse is equal to R + S = 49 + 64 = 113 cm.The hypotenuse is equal to the square root of the number 113.

Angles of a right triangle

The result was an unreasonable number.

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 Pyghagorean triplet, since they satisfy the relation x? + Y? = Z, which is natural.

Other examples of the 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, suppose such a hand is 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 do not need A.

fifth Pythagorean theorem - special case, which is more of 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 to the 90 degree angle.

instructions

first In the case of known catheters, as well as an acute angle of a right-angled triangle, the hypotenuse size may equal 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 C2?) / Cos?. Example: Let ABC be an irregular triangle with hypotenuse AB and right angle C.

Let B be 60 degrees and A 30 degrees. BC stem length 8 cm. AB hypotenuse length should be found. To do this, you can use one of the above methods: AB = BC / cos60 = 8 cm. AB = BC / sin30 = 8 cm.

Hypotenuse is the longest side of the rectangle triangle... It is located at right angles. Rectangle hypotenuse search method 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 of nego is located. the hypotenuse of the leg at the cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio of the sinusoidal angles: da = a / sin.

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An angular triangle, the sides of which 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 with an angle of 90 °. The side opposite to the right corner is called the hypotenuse, the other side is called the legs.

If you want to find how a right-angled triangle is formed by some of the 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 corners are the same.

To calculate the angle of a right-angled equal triangle, you need to know that:

• This is no worse than 90 °.
• Acute angle values ​​are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.

  The angles α and β are equal to 45 °.

If the known value of one of the acute angles is known, the other can be found by the formula: β = 180º-90º-α or α = 180º-90º-β.

This ratio is most often used when one of the angles is 60 ° or 30 °.

Key concepts

The sum of the interior angles of a triangle is 180 °.

Because this is one level, two remain sharp.

Calculate triangle online

If you want to find them, you need to know that:

other methods

The acute angle values ​​of a right-angled 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 dropped 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:

• 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 pages

If the lengths of the hypotenuse and one of the legs are known in a right-angled triangle or on both 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 from its side.

Perimeter of an Equal Triangle can be found by concatenating the side lengths sequentially, or by multiplying the side length by 2 and adding the base length 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 is the base.

Perimeter of an equilateral triangle can be found by sequentially concatenating the length of its sides or by multiplying the length of any page by 3.

The general formula for finding the rim of equilateral triangles will 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 the 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 of the parallelogram range.

Since the area of ​​the parallelogram is the same as the product of its base height, the area of ​​the triangle will be half that product. Thus, for ΔABC, the region will be the same

Now consider a right-angled triangle:

Two identical right-angled triangles can be bent into a rectangle if it leans against them, which is each 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-angled triangle is equal to the product of the legs, divided by 2.

From these examples, it can be inferred that the surface of each triangle is the same as the product of the length, and the height is reduced to a substrate 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.

Defining a triangle

Triangle- this is geometric figure, which is formed as a result of the intersection of three line segments, the ends of which do not lie on one straight line. Any triangle has three sides, three vertices, and three corners.

Online calculator

Triangles are of various kinds. For example, there is an equilateral triangle (one in which all sides are equal), isosceles (two sides are equal in it) and right-angled (in which one of the corners is straight, that is, equal to 90 degrees).

The area of ​​a triangle can be found in various ways, depending on which elements of the figure are known by the condition of the problem, be it the angles, lengths, or, in general, the radii of the circles associated with the triangle. Let's consider each method separately with examples.

Formula for the area of ​​a triangle by base and height

S = 1 2 ⋅ a ⋅ h S = \ frac (1) (2) \ cdot a \ cdot hS =2 1 ​ ⋅ a ⋅h,

A a a- the base of the triangle;
h h h- the height of the triangle drawn to the given base a.

Example

Find the area of ​​a triangle if the length of its base is known, equal to 10 (cm.) And the height drawn to this base, equal to 5 (cm.).

Solution

A = 10 a = 10 a =1 0
h = 5 h = 5 h =5

We substitute in the formula for the area and we get:
S = 1 2 ⋅ 10 ⋅ 5 = 25 S = \ frac (1) (2) \ cdot10 \ cdot 5 = 25S =2 1 ​ ⋅ 1 0 ⋅ 5 = 2 5 (see sq.)

Answer: 25 (cm.)

The formula for the area of ​​a triangle by the lengths of all sides

S = p ⋅ (p - a) ⋅ (p - b) ⋅ (p - c) S = \ sqrt (p \ cdot (p-a) \ cdot (p-b) \ cdot (p-c))S =p ⋅ (p - a) ⋅ (p - b) ⋅ (p - c)​ ,

A, b, c a, b, c a, b, c- the lengths of the sides of the triangle;
p p p- half the sum of all sides of the triangle (that is, half the perimeter of the triangle):

P = 1 2 (a + b + c) p = \ frac (1) (2) (a + b + c)p =2 1 ​ (a +b +c)

This formula is called Heron's formula.

Example

Find the area of ​​a triangle if the lengths of its three sides are known, equal to 3 (see), 4 (see), 5 (see).

Solution

A = 3 a = 3 a =3
b = 4 b = 4 b =4
c = 5 c = 5 c =5

Find half of the perimeter p p p:

P = 1 2 (3 + 4 + 5) = 1 2 ⋅ 12 = 6 p = \ frac (1) (2) (3 + 4 + 5) = \ frac (1) (2) \ cdot 12 = 6p =2 1 ​ (3 + 4 + 5 ) = 2 1 ​ ⋅ 1 2 = 6

Then, according to Heron's formula, the area of ​​a triangle is:

S = 6 ⋅ (6 - 3) ⋅ (6 - 4) ⋅ (6 - 5) = 36 = 6 S = \ sqrt (6 \ cdot (6-3) \ cdot (6-4) \ cdot (6- 5)) = \ sqrt (36) = 6S =6 ⋅ (6 − 3 ) ⋅ (6 − 4 ) ⋅ (6 − 5 ) ​ = 3 6 ​ = 6 (see sq.)

Answer: 6 (see sq.)

Formula for the area of ​​a triangle on one side and two angles

S = a 2 2 ⋅ sin ⁡ β sin ⁡ γ sin ⁡ (β + γ) S = \ frac (a ^ 2) (2) \ cdot \ frac (\ sin (\ beta) \ sin (\ gamma)) ( \ sin (\ beta + \ gamma))S =2 a 2 ​ ⋅ sin (β + γ)sin β sin γ ​ ,

A a a- the length of the side of the triangle;
β, γ \ beta, \ gamma β , γ - corners adjacent to the side a a a.

Example

Given a side of a triangle equal to 10 (see) and two adjacent angles of 30 degrees. Find the area of ​​a triangle.

Solution

A = 10 a = 10 a =1 0
β = 3 0 ∘ \ beta = 30 ^ (\ circ)β = 3 0 ∘
γ = 3 0 ∘ \ gamma = 30 ^ (\ circ)γ = 3 0 ∘

According to the formula:

S = 1 0 2 2 ⋅ sin ⁡ 3 0 ∘ sin ⁡ 3 0 ∘ sin ⁡ (3 0 ∘ + 3 0 ∘) = 50 ⋅ 1 2 3 ≈ 14.4 S = \ frac (10 ^ 2) (2) \ cdot \ frac (\ sin (30 ^ (\ circ)) \ sin (30 ^ (\ circ))) (\ sin (30 ^ (\ circ) +30 ^ (\ circ))) = 50 \ cdot \ frac ( 1) (2 \ sqrt (3)) \ approx14.4S =2 1 0 2 ​ ⋅ sin (3 0 ∘ + 3 0 ∘ ) sin 3 0 ∘ sin 3 0 ∘ ​ = 5 0 ⋅ 2 3 ​ 1 ​ ≈ 1 4 . 4 (see sq.)

Answer: 14.4 (cm.)

The formula for the area of ​​a triangle on three sides and the radius of the circumscribed circle

S = a ⋅ b ⋅ c 4 R S = \ frac (a \ cdot b \ cdot c) (4R)S =4 Ra ⋅ b ⋅ c​ ,

A, b, c a, b, c a, b, c- sides of the triangle;
R R R- the radius of the circumscribed circle around the triangle.

Example

We take the numbers from our second problem and add the radius to them R R R circles. Let it be equal to 10 (see).

Solution

A = 3 a = 3 a =3
b = 4 b = 4 b =4
c = 5 c = 5 c =5
R = 10 R = 10 R =1 0

S = 3 ⋅ 4 ⋅ 5 4 ⋅ 10 = 60 40 = 1.5 S = \ frac (3 \ cdot 4 \ cdot 5) (4 \ cdot 10) = \ frac (60) (40) = 1.5S =4 ⋅ 1 0 3 ⋅ 4 ⋅ 5 ​ = 4 0 6 0 ​ = 1 . 5 (see sq.)

Answer: 1.5 (see sq.)

The formula for the area of ​​a triangle on three sides and the radius of the inscribed circle

S = p ⋅ r S = p \ cdot rS= p⋅ r,

p pp- half of the perimeter of the triangle:

p = a + b + c 2 p = \ frac (a + b + c) (2)p= 2 a+ b+ c​ ,

a, b, c a, b, ca, b, c- sides of the triangle;
r rr is the radius of a circle inscribed in a triangle.

Example

Let the radius of the inscribed circle be 2 (see). We take the lengths of the sides from the previous problem.

Solution

$a=3b\; =\; 4\; b\; =\; 4b=4c\; =\; 5\; c\; =\; 5c=5r\; =\; 2\; r\; =\; 2r=2$

$p=23+4+5=6$

S = 6 ⋅ 2 = 12 S = 6 \ cdot 2 = 12S= 6 ⋅ 2 = 1 2 (see sq.)

Answer: 12 (see apt.)

The formula for the area of ​​a triangle on two sides and the angle between them

S = 1 2 ⋅ b ⋅ c ⋅ sin ⁡ (α) S = \ frac (1) (2) \ cdot b \ cdot c \ cdot \ sin (\ alpha)S= 2 1 ​ ⋅ b⋅ c⋅ sin(α ) ,

b, c b, cb, c- sides of the triangle;

α \ alphaα - the angle between the sides b bb and c cc.

Example

The sides of the triangle are 5 (see) and 6 (see), the angle between them is 30 degrees. Find the area of ​​a triangle.

Solution

$b=5c\; =\; 6\; c\; =\; 6c=6\alpha \; =\; 3\; 0\; \circ \; \backslash \; alpha\; =\; 30\; ^\; (\backslash \; circ)\alpha =30^{\circ }$

S = 1 2 ⋅ 5 ⋅ 6 ⋅ sin ⁡ (3 0 ∘) = 7.5 S = \ frac (1) (2) \ cdot 5 \ cdot 6 \ cdot \ sin (30 ^ (\ circ)) = 7.5S= 2 1 ​ ⋅ 5 ⋅ 6 ⋅ sin(3 0 ∘ ) = 7 . 5 (see sq.)

Answer: 7.5 (cm.)

The first are the segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the 90 ° angle. A Pythagorean triangle is one whose sides are equal to natural numbers; their lengths in this case are called "Pythagorean triplets".

Egyptian triangle

In order for the current generation to learn geometry in the form in which it is taught at school now, it has developed for several centuries. The fundamental point is considered the Pythagorean theorem. The sides of the rectangular are known all over the 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) = 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". The interesting thing is that which is inscribed in the figure is equal to one. The name originated around the 5th century BC, when Greek philosophers traveled to Egypt.

When building the pyramids, architects and surveyors used a ratio of 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 shapes

• An acute angle in a right-angled triangle and a large side, which are equal to the same elements in the second triangle, are an indisputable sign of equality of 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 the same in the second characteristic.
• When two figures are superimposed on each other, we rotate them so that, when combined, they become one isosceles triangle... By its property, the sides, or rather, the hypotenuses, are equal, as are the angles at the base, which means that these figures are the same.

On the first basis, 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 in sign II, the essence of which is the equality of the leg and the acute angle.

Right Angle Triangle Properties

The height dropped from the right angle splits the figure into two equal parts.

The sides of a right-angled triangle and its median are easy to recognize by the rule: the median, which is lowered by the hypotenuse, is equal to its half. 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-angled triangle, the properties of angles of 30 °, 45 ° and 60 ° apply.

• At an angle of 30 °, it should be remembered that opposite leg will be equal to 1/2 of the largest side.
• If the angle is 45 °, then the second acute angle is also 45 °. This suggests that the triangle is isosceles, and its legs are the same.
• The property of a 60 ° angle is that the third angle has a degree measure of 30 °.

The area can be easily recognized by one of three formulas:

1. through the height and side to which it descends;
2. according to Heron's formula;
3. on the sides and the corner between them.

The sides of a right-angled triangle, or rather the legs, converge at two heights. In order to find the third, it is necessary to consider the resulting triangle, and then, by the Pythagorean theorem, calculate the required length. In addition to this formula, there is also the ratio of the doubled area and the length of the hypotenuse. The most common expression among students is the former, as it requires less calculations.

Theorems applied to a right triangle

The geometry of a right triangle includes the use of theorems such as:

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, area of ​​the slopes, etc. How to calculate the angle of inclination of the roof correctly? It is on 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 if it comes about the roof of a residential building or a roof with a complex shape. But even an ordinary single-slope one, 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, do not find out which optimal height must have a ridge, then there is a great risk of building such a roof that will collapse after the first snowfall, or the entire finishing coating will be torn off from it even by a moderately strong 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 ridge

Roofing ridge

Units

Remembering the geometry that everyone studied in school, it is safe to say that the angle of inclination 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 of the slope is the angle that is formed by two intersecting planes- overlapping and directly with a roof slope. It can only be sharp, that is, lie in the range of 0-90 degrees.

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 the angles of the roof in degrees, everything is simple - everyone who has studied geometry at school has this knowledge. It is enough to sketch a roofing diagram on paper and use a protractor to determine the angle.

As for the percentage, 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, the usual tilt is 2.22%. That is, a slope with an angle of 45 normal degrees is 100%. And 1 percent is 27 arc minutes.

Values ​​table - 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 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. At one point, they can play their part. Determine the appropriate angle of inclination of the roof, knowing:

• the types of materials from which the roofing pie will be built, starting from the rafter system and ending with external decoration;
• climate conditions in a given area (wind load, prevailing wind direction, amount of precipitation, etc.);
• the shape of the future structure, its height, design;
• the purpose of the structure, options for using the attic space.

In 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 resisting 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 the 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 make it possible not to take into account the snow load in general, but the wind pressure in this case on the roof will be 5 times more than on the roof with a slope of only 11 degrees.

On a note! The more the parameters of the slope of the roof, the more materials will be required to create it. The cost increases by at least 20%.

Slope corners 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. Optimal slope angles for roofs made of various materials.

On a note! The lower the slope of the roof, the smaller the step is used when creating the lathing.

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Metal tile

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 upper point, that is, in the ridge or the transition of the lower part of the entire rafter system to the upper one (in the case of attic roofs), as well as the projection of the length of a particular slope onto the horizontal, which is represented by the slabs. 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 section will vary depending on the angle of inclination.

Knowledge of the formulas from trigonometry will help to design the roof: tgA = H / L, sinA = H / S, H = LхtgA, S = H / sinA, where A is the slope angle, H is the height of the roof to the ridge area, L is ½ of the entire length span of the roof (with a gable roof) or the entire length (in the case of a pitched roof), S is 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 calculation is based on the roof angle, 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.

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

Triangles are isosceles, equilateral, and non-sided. Of 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 called the hypotenuse).

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The 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-angled triangle is equal to the square of its hypotenuse: a² + b² = c²

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

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. This requires a third parameter: it can be the angle, the height of the area of ​​the triangle, the radius of the circle inscribed in it, etc.

If the perimeter is known

In this case, the task is even easier. 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 transferring all known parameters in one direction from the equal sign:

2) Substitute the values ​​instead 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 by the angle and two other sides, the solution is reduced to calculating the 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 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 cannot be dispensed with.

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

sin γ = 2S / (a ​​* b)

2) Using the following formula, we 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 get the answer to the problem.