How to calculate how many degrees angle. Triangle side. Examples from real life
The triangle is a geometric number consisting of three segments that connect three points that are not lying on the same line. Points that form a triangle are called its points, and the segments 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 source data and the conditions of the problem.
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To calculate the sides of the rectangular triangle, the Pythagora theorem is used, according to which the square of the hypotenuse is equal to the sum of the squares of the leg.
If we celebrate the legs of the letters "a" and "b", and hypotenuse - "C", then pages can be found with the following formulas:
If the sharp corners of the rectangular triangle (A and B) are known, it can be found with the following formulas:
Cropped triangle
The triangle is called an equilateral triangle, in which both sides are the same.
How to find hypotenuse in two legs
If the letter "A" is identical to the same page, "B" - the base, "B" - an angle opposite to the base, "A" - the adjacent angle for calculating pages can use the following formulas:
Two corners and lateral
If one page (C) and two angle (a and b) of any triangle are known, the sinus formula is used to calculate the remaining pages:
You must find the third value y \u003d 180 - (A + B), because
the sum of all corners of the triangle is 180 °; 
Two sides and angle
If two sides of the triangle (a and b) are known and the angle between them (y), the cosine theorem can be used to calculate a third party.
How to determine the perimeter of a rectangular triangle
The triangular triangle is a triangle, one of which is 90 degrees, and the other two are sharp. payment perimeter such triangle Depending on the number of well-known information about this.
You need it
- Depending on the case, the skills of 2 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, independently, perpendicular or not triangular, the perimeter is calculated as: p \u003d a + b + c, where it is possible, C - hypotenuse; A and B - legs.
second Method 2.
If there are only two sides in a rectangle, then using Pythagore's theorem, triangle It can be calculated by the formula: p \u003d V (A2 + B2) + A + B or P \u003d V (C2 - B2) + B + C.
the third Method 3. Let the hypotenuse C and the sharp angle? Considering the rectangular triangle, it will be possible to detect the perimeter in this way: p \u003d (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 by the formula: P \u003d A * (1 / TG?
1 / son? + 1)
fifty Method 5.
Online calculation of the triangle
Allow our leg to give and be included in it, then the range will be calculated as: P \u003d A * (1 / CTG + 1 / + 1 COS?)
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Pythagoreo Theorem is the basis of any mathematics. Determines the relationship between the parties of the true triangle. It is now indicated 367 evidence of this theorem.
instructions
first The classic school formulation of the Pythagoreo theorem sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs.
To find hypotenuse in rectangular triangle two catts, you should turn to build a square foot length, collect them and take square root From the amount. In the original wording of his statement, the market is based on hypotenuse equal to the sum of squares of 2 squares production of Catete. However, modern algebraic formulation does not require the introduction of the representation of the region.
second For example, a rectangular triangle whose legs are 7 cm and 8 cm.
Then, according to the Pythagora theorem, the square hypotenuse is R + S \u003d 49 + 64 \u003d 113 cm. The hypotenuse is equal to the square root from among the 113.
Corners of a rectangular triangle
The result was the unreasonable number.
the third If triangles are legs 3 and 4, then hypotenuse \u003d 25 \u003d 5. When you retake the square root, you get a natural number. Numbers 3, 4, 5 form a Pigagorean triplet, since they satisfy the X ratio? + Y? \u003d 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 Pythagora theorem turns into a more primitive equation. For example, let such a hand be equal to number A and hypotenuse defined for C and then with? \u003d Ap + Ap, C \u003d 2a2, C \u003d a? 2. In this case, you do not need A.
fifty Pythagorean theorem - private casewhich is greater than the general cosine theorem, which establishes the connection between the three sides of the triangle for any angle between the two of them.
Tip 2: How to Determine Hypotenuse for Foot and Corners
The hypotenuse is called the side in a rectangular triangle, which is opposite to the corner of 90 degrees.
instructions
first In the case of known catheters, as well as the acute angle of the rectangular triangle can hypotenuses, equal to the ratio to the cosine / sine of this angle, if the angle was opposite / e include: H \u003d C1 (or C2) / sin, H \u003d C1 (or C2?) / Cos? Example: Let ABC be given an irregular triangle with hypothenoise AB and at a right angle C.
Let B be equal to 60 degrees and a 30 degrees. Length of the legs BC 8 cm. The length of the AB hypotenuse should be detected. To do this, you can use one of the above methods: AB \u003d BC / COS60 \u003d 8 cm. AB \u003d BC / SIN30 \u003d 8 cm.
Hypotenuse - the longest side of the rectangle triangle . It is located at right angles. Rectangle Hypotenuse triangle Depending on the source data.
instructions
first If your legs are perpendicular triangle , then the length of the rectangle hypotenuse triangle It can be detected by a Pythagorean analogue - the square of the hypotenuse length is equal to the sum of the squares of the leg lengths: C2 \u003d A2 + B2, where a and b - the length of the legs is right triangle .
second If one of the legs under an acute angle, the formula for finding the hypotenuse will depend on the presence or absence of a certain angle relative to the well-known cathelet - adjacent (cathet is located near), or vice versa (the opposite case of the NEGO.v of the specified angle is equal to the share Leg hypotenuse in a cosine angle: a \u003d a / cos; E, on the other hand, hypotenuse is the same as the ratio of sinusoidal angles: Da \u003d A / SIN.
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Useful advice
The angular triangle whose side is associated as 3: 4: 5, called the Egyptian delta, due to the fact that these figures are widely used by architects of ancient Egypt.
This is also the easiest example of Jeron's triangles, in which pages and areas are represented by integers.
The triangle is called a rectangle, the angle of which is 90 °. The side opposite to the right corner is called hypotenuse, the other - legs.
If you want to find a rectangular triangle formed by some properties of the right triangles, namely the fact that the amount of sharp 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 angle is the same.
To calculate the angle of a rectangular equal triangle, you need to know that:
- It is no worse than 90 °.
- The values \u200b\u200bof acute angles are determined by the formula: (180 ° - 90 °) / 2 \u003d 45 °, i.e.
The angles α and β are 45 °.
If the known value of one of the sharp corners is known, the other can be found according to the formula: β \u003d 180º-90º-α or α \u003d 180º-90º-β.
This ratio is most often used if one of the corners is 60 ° or 30 °.
Key concepts
The sum of the internal angles of the triangle is 180 °.
Because it is one level, two remain sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The sharp corners of the rectangular triangle can be calculated from the average value - with the line from the point on the opposite side of the triangle, and the height - the line is a perpendicular, lowered from the hypotenuse at right angles.
Let the median stretches from the right angle to the middle of the hypotenuse, and H is height. In this case it turns out that: 
- sin α \u003d b / (2 * s); SIN β \u003d A / (2 * S).
- cos α \u003d a / (2 * s); cos β \u003d b / (2 * s).
- sin α \u003d h / b; sin β \u003d h / a.
Two pages
If the lengths of hypotenuses and one of the legs are known in the rectangular triangle or on both sides, trigonometric identities are used to determine the values \u200b\u200bof sharp corners:
- α \u003d arcsin (A / C), β \u003d Arcsin (B / C).
- α \u003d arcos (b / c), β \u003d arcos (A / C).
- α \u003d arctg (A / B), β \u003d arctg (b / a).
The length of the rectangular triangle
Square and triangle square
perimeter
The circumference of any triangle is equal to the sum of the length of the three sides. General formula for finding a triangular triangle:
where P is the circumference of the triangle, A, B and C of his side.
Perimeter of equal triangle It can be found by a sequential combination of lengths of its sides or multiplying the side length of 2 and adding to the product of the base length.
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 basis.
Perimeter of equilateral triangle It can be found by a consistent combination of the length of its parties or by multiplying the length of any page to 3.
The general formula for finding the rim of equilateral triangles will look like this:
where P is the perimeter of the equilateral triangle, A is any of its sides.
region
If you want to measure the triangle area, you can compare it with a parallelogram. Consider the ABC triangle:
If we take the same triangle and fix it so that we get a parallelogram, we will get the parallelograms of the same height and the foundation as this triangle:
In this case, the overall side of the triangles is folded together on the diagonal of the molded parallelogram.
From the properties of the parallelogram. It is known that the diagonal of the parallelogram is always divided into two equal triangles, the surface of each triangle is equal to half the range of the parallelogram.
Since the parallelogram area coincides with the product of its base height, the triangle area will be equal to half of this product. Thus, for ΔABC region will be the same
Now consider the rectangular triangle:
Two identical rectangular triangles can be bend into a rectangle if it leans them that every other hypotenuse.
Since the surface of the rectangle coincides with the surface of the neighboring sides, the area of \u200b\u200bthis triangle is the same:
From here you can conclude that the surface of any rectangular triangle is equal to the work of the legs divided by 2.
From these examples, it can be concluded that the surface of each triangle is the same as the product of length, and the height is reduced to the substrate divided by 2.
The general formula for finding a triangle area will look like this:
where S is the region of the triangle, but its base, but height falls to the bottom a.
The first are segments that fit to a straight corner, and the hypotenuse is the longest part of the figure and is opposite the angle in 90 o. Pythagora triangle is called the part of which are equal to natural numbers; Their lengths in this case are called "Pytagorova Troika".
Egyptian triangle
In order for the current generation to find out the geometry in the form in which it is taught at school now, it developed several centuries. The fundamental point is considered the Pythagora theorem. The sides of the rectangular are known for the whole world) make up 3, 4, 5.
Few people are not familiar with the phrase "Pythagoras pants in all directions are equal." 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 cathets).
Among mathematicians, a triangle with the parties 3, 4, 5 (see, m, etc.) is called "Egyptian". Interestingly, which is inscribed in the figure, is equal to one. The name arose about the V century BC, when Greece philosophers went to Egypt.
When building pyramids, architects and land surveyors used the 3: 4: 5 ratio. Such structures were obtained proportional, pleasant in appearance and spacious, and also rarely collapsed.
In order to build a straight corner, the builders used the rope on which 12 nodes were tied. In this case, the probability of building a rectangular triangle rose to 95%.
Signs of equality of figures
- Acute angle in a rectangular triangle and a large side that are equal to the same elements in the second triangle, is an indisputable sign of equality of figures. Taking into account the amount of corners, it is easy to prove that the second sharp corners are also equal. Thus, triangles are the same on the second basis.
- When you apply two figures on each other, they will turn them in such a way that they, having shared, become a single-equated triangle. According to its function, the parties, or rather, hypotenuses are equal, as well as the angles at the base, and therefore these figures are the same.
On the first sign, it is very easy to prove that the triangles are really equal, the main thing is that two smaller sides (i.e. kartets) were equal to each other.
Triangles will be the same in the II sign, the essence of which is the equality of catech and acute angle.
Triangle Properties with Direct Angle
The height that was lowered from the straight angle, breaks the figure into two equal parts.
The sides of the rectangular triangle and its medians are easy to learn according to the rule: the median, which is lowered on the hypotenuse, is equal to her half. It can be found both according to the formula of Geron and according to the statement that it is equal to half the work of cathets.
In a rectangular triangle, there are properties of angles in 30 o, 45 o and 60 o.
- At an angle, which is 30 o, should be remembered that anti-cathet It will be 1/2 of the largest side.
- If the angle is 45 oh, then the second sharp angle is also 45 o. This suggests that the triangle is a preceded, and his cathets are the same.
- The body of an angle of 60 o is that the third angle has a degree measure in 30 o.
The area is easy to find out for one of three formulas:
- through the height and side to which it goes;
- according to the formula of Geron;
- on the sides and the corner between them.
The sides of the rectangular triangle, or rather kartettes, converge with two heights. In order to find the third, it is necessary to consider the resulting triangle, and then according to the Pythagora theorem, calculate the necessary length. In addition to this formula, there is also the ratio of the double area and the length of the hypotenuse. The most common expression among students is the first, since it requires less calculations.
Theorems applied to a rectangular triangle
The geometry of the rectangular triangle includes the use of such theorems as:
Build any roof is not as simple as it seems. And if you want it to be reliable, durable and not afraid of various loads, then pre-at once at the design stage, you need to make a lot of calculations. And they will include not only the number of materials used for installation, but also determining the angles of inclination, rope area, etc. How to calculate the angle of inclination of the roof correctly? It is from this value that the remaining parameters of this design will depend largely.
Design and construction of any roof - always very important and responsible. Especially if we are talking About the roof of a residential building or complex on the shape of the roof. But even the usual single-sided, installed on an unpleasant shed or garage, the same needs preliminary calculations.
If you do not define the angle of inclination in advance, do not find out what optimum height Must have a horse, the risk of constructing such a roof, which will collapse after the first snowfall, or all the finishing coating from it will be torn even a moderate strength of the wind.
Also, the angle of inclination of the roof will significantly affect the height of the skate, on the area and dimensions of the skates. Depending on this, it will be possible to more accurately calculate the number of materials required to create a rafter system and finishing materials.
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Roofing horse
Units
Remembering the geometry, which everyone studied at school, can with confidence to declare that the angle of inclination of the roof is measured in degrees. However, in the books dedicated to construction, as well as in various drawings, you can meet another option - the angle is specified as a percentage (here is due to the aspect ratio).
Generally, the angle of inclination of the skate is an angle that is formed by two intersecting planes. - overlapping and directly with the roof. It can only be sharp, that is, lying in the range of 0-90 degrees.
On a note! Very steep slides, the angle of inclination of which is more than 50 degrees, are extremely rare in its pure form. Usually they are used only with decorative design of the roofs, may be present in the attic.
As for the measurement of the corners of the roof in degrees, then everything is simple - everyone who studied in school geometry has this knowledge. It is enough to draw a roof circuit on paper and using the transporter to determine the angle.
As for interest, then it is necessary to know the height of the skate and width of the building. The first indicator is divided into the second, and the obtained value is multiplied by 100%. Thus, one can calculate the percentage ratio.
On a note! With percentage ratio of 1, the usual degree inclination is 2.22%. That is, the skate with an angle of 45 ordinary degrees is 100%. A 1 percent is 27 angular minutes.
Table of values \u200b\u200b- degrees, minutes, interest
What factors affect the angle of inclination?
At the angle of inclination, any roof is influenced by a very large number of factors, starting 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 role. To determine the appropriate angle of inclination of the roof should be known:
- types of materials from which the roofing pie will be built, ranging from the rafter system and ending with the outer finish;
- climate conditions in this area (wind load, the predominant direction of winds, the amount of precipitation, etc.);
- the form of the future structure, its height, design;
- purpose of the structure, options for using an attic room.
In those regions where strong wind load is noted, it is recommended to build a roof with one row and a small angle of inclination. Then, with a strong wind, the roof had more chances to resist and not be torn. If the region is characterized by a large amount of rainfall (snow or rain), then the slope is better to do sharp - this will allow the sediments to roll / drain from the roof and not to create additional loads. The optimal slope of a single-table roof in windy regions varies within 9-20 degrees, and where many precipitation falls - up to 60 degrees. The angle of 45 degrees will allow not to take into account the snow load in general, but the wind pressure in this case will be 5 times more on the roof than on the roof with a slope of only 11 degrees.
On a note! The greater the roof slope parameters, the greater the amount of materials will be required to create it. The cost increases at least 20%.
Corners of rods and roofing materials
Not only climatic conditions will have a significant impact on the shape and angle of the skates. An important role is played by materials used for construction, in particular - roof covering.
Table. Optimal angles of slopes for roofs from various materials.
On a note! The smaller the inclination of the roof, the smaller step is used when creating a crate.
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Metal tile.
The height of the skate also depends on the corner of the skate
When calculating any roof behind the landmark, a rectangular triangle is always taken, where kartets are the height of the skate at the top point, that is, in the skate or the transition of the lower part of the entire system has rafted into the upper (in the case of mansard roofs), as well as the projection of the length of a specific slide on Horizontal, which is represented by overlaps. There is only one constant value here is the length of the roof between the two walls, that is, the length of the span. The height of the skate part will vary depending on the angle of inclination.
Designing the root will help knowledge of formulas from trigonometry: TGA \u003d H / L, SINA \u003d H / S, H \u003d LXTGA, S \u003d H / SINA, where A is the angle of the skate, H is the height of the roof to the skate region, L - ½ of the total length The span of the roof (with a double roof) or the entire length (in the case of a single roof), S is the length of the row itself. For example, if the exact value of the height of the skunk part is known, the angle of inclination is determined by the first formula. It will be possible to find an angle on the Tangent table. If the basis of the calculations is the roof angle, then it is possible to find the height parameter of the skate in the third formula. The length of the rafter, having an angle value of the inclination and parameters of cathets, can be calculated in the fourth formula.
The rectangular triangle is found in reality 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 to solve problems according to geometry, but also in life situations.
Geometry triangles
In the elementary geometry, the rectangular triangle is a figure that consists of three connected segments forming three angle (two sharp and one straight). The rectangular triangle is an original figure characterized by a number of important properties that make up the foundation of trigonometry. Unlike a conventional triangle, the side of the rectangular figure have their own names:
- Hypotenuse is the longest side of the triangle, lying opposite the direct angle.
- Kartets - segments that form a straight corner. Depending on the angle under consideration, the catat can be adjacent to it (forming this angle with hypotenuse) or opposite (lying opposite the angle). For indirectant triangles of cathets do not exist.
It is the ratio of cathets and hypotenuses that makes the basis of trigonometry: sinuses, tangents and sedes are defined as the ratio of the sides of the rectangular triangle.
Rectangular triangle in reality
This figure was widespread in reality. Triangles are used in designing and technology, so the calculation of the area of \u200b\u200bthe figure has to be carried out by engineers, architects and designers. The shape of the triangle has the bases of tetrahedra or prisms - three-dimensional figures that are easy to meet in everyday life. In addition, the square is the simplest representation of the "flat" rectangular triangle in reality. The square is a plumbing, drawing, construction and joiner tool, which is used to build corners as schoolchildren and engineers.
Area of \u200b\u200ba triangle
Area geometric Figure - This is a quantitative assessment of which part of the plane is limited by the sides of the triangle. The area of \u200b\u200ban ordinary triangle can be found five ways using the Geron formula or operating when calculating with such variables as the base, side, angle and radius of the inscribed or described circle. The simplest formula of the square is expressed as:
where a is the side of the triangle, H is its height.
The formula for calculating the area of \u200b\u200bthe rectangular triangle is even easier:
where a and b - kartets.
Working with our online calculator, you can calculate the triangle area using three pairs of parameters:
- two categories;
- catat and prudent corner;
- root and opposite corner.
In tasks or household situations, you will be given different combinations of variables, so this form of the calculator allows you to calculate the triangle area in several ways. Consider a couple of examples.
Examples from real life
Ceramic tile
Suppose you want to complete the walls of the kitchen with ceramic tiles, which has the shape of a rectangular triangle. In order to determine the tile consumption you must find out the area of \u200b\u200bone facing element and the total area of \u200b\u200bthe treated surface. Let you need to handle 7 square meters. The length of the cathets of one element is 19 cm, then the tile area will be equal to:
This means that the area of \u200b\u200bone element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that for finishing 7 square meters of the wall you will need 7 / 0.01805 \u003d 387 elements of the facing tile.
School task
Suppose in the school challenge of geometry, it is required to find the area of \u200b\u200ba rectangular triangle, knowing only that the side of one category is 5 cm, and the value of the opposite angle is 30 degrees. Our online calculator is accompanied by an illustration on which the parties and the angles of the rectangular triangle are indicated. If the side A \u003d 5 cm, then its opposite angle is an angle of 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 \u200b\u200ba given triangle, but also determines the length of the adjacent catech and hypotenuse, as well as the value of the second angle.
Conclusion
Rectangular triangles are found in our lives literally at every corner. The definition of such figures is useful to you not only when solving school tasks on geometry, but also everyday and professional activities.
Rectangular is called a triangle, one of the angles of which is 90º. The side, opposing the straight corner, is called hypotenuse, and two others - by customs.
To find an angle in a rectangular triangle, some properties of rectangular triangles are used, namely: the fact that the sum of sharp corners is 90º, as well as that opposite the category, the length of which is twice as fewer hypotenuse, is an angle equal to 30º.
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Isosceles triangle
One of the properties equal triangle - Two corners are equal. To calculate the values \u200b\u200bof the angles of a rectangular isceived triangle, you need to know that:
- The straight angle is 90º.
- The values \u200b\u200bof acute angles are determined by the formula: (180º-90º) / 2 \u003d 45º, i.e. The angles α and β are equal to 45º.
If a value of one of the sharp corners is known, the second can be found according to the formula: β \u003d 180º-90º-α, or α \u003d 180º-90º-β. Most often, this ratio is used if one of the corners is 60º or 30º.
Key concepts
The sum of the internal corners of the triangle is 180º. Since one corner of the direct, the two remaining will be sharp. To find them you need to know that:
other methods
The magnitudes of the sharp corners of the rectangular triangle can be calculated, knowing the median value of the line, carried out from the vertex to the opposite side of the triangle, and the height is a direct, representing a perpendicular, lowered from a direct angle to the hypotenuse. Let S be a median spent from a direct angle to the middle of the hypotenuse, H is height. In this case, it turns out that: 
- sin α \u003d b / (2 * s); SIN β \u003d A / (2 * S).
- cos α \u003d a / (2 * s); cos β \u003d b / (2 * s).
- sin α \u003d h / b; sin β \u003d h / a.
Two sides
If the length of hypotenuses and one of the nuts are known in the rectangular triangle, or two sides, trigonometric identities are used to find the values \u200b\u200bof sharp corners:
- α \u003d arcsin (A / C), β \u003d Arcsin (B / C).
- α \u003d arcos (b / c), β \u003d arcos (A / C).
- α \u003d arctg (A / B), β \u003d arctg (b / a).
