Online calculator.
Solution of triangles.

The solution of a triangle is the finding of all its six elements (i.e., three sides and three angles) by any three given elements that define the triangle.

This math program finds side \(c \), angles \(\alpha \) and \(\beta \) given 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 can be useful for high school students in preparing for tests and exams, when testing knowledge before the Unified State Examination, and 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 your math or algebra homework done as quickly as possible? In this case, you can also use our programs with a detailed solution.

In this way, you can conduct your own training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.

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 set not only whole, but also fractional.
The integer and fractional parts in decimal fractions can be separated by either a dot or a comma.
For example, you can enter decimals like 2.5 or like 2.5

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

Because There are a lot of people who want to solve the problem, your request is queued.
After a few seconds, the solution will appear below.
Wait, please sec...

If you noticed an error in the solution, then you can write about it in the Feedback Form .
Do not forget indicate which task you decide what enter in the fields.

Our games, puzzles, emulators:

A bit of theory.

Sine theorem

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 in triangle ABC AB = c, BC = a, CA = b. Then
The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the angle between them.
$$ a^2 = b^2+c^2-2ba \cos A $$

Solving Triangles

The solution of a triangle is the finding of all its six elements (i.e., three sides and three angles) by any three given elements that define the triangle.

Consider three problems for solving a triangle. In this case, we will use the following notation for the sides of the triangle ABC: AB = c, BC = a, CA = b.

Solution of a triangle given two sides and an angle between them

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

Solution
1. By the law of cosines we find \(c\):

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

Solution of a triangle given a side and adjacent angles

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

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

Solving a Triangle with Three Sides

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

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

2. Similarly, we find the 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. By the sine theorem 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:
$$ c = a \frac(\sin C)(\sin A) $$

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 the angle of inclination of the roof in advance, do not find out what optimal height the ridge should have, then there is a high risk of building a roof that will collapse after the first snowfall, or all the finishing coating will be torn off from it 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 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.

Quick navigation for an article

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. calculation 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

the 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.

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 is 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?)

Similar videos

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

the 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.

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

the first In the case of well-known catheters, as well as an acute angle of a right triangle, the hypotenuse can have a size equal to the ratio of the leg to the cosine / sine of this angle, if the angle was opposite / e include: H \u003d C1 (or C2) / sin, H \u003d 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

the 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.

Similar videos

Helpful Hints
An angular 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°.

Quick navigation for an article

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.

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 the legs.

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

Quick article navigation

Isosceles triangle

One of the properties of an isosceles triangle is that two of its angles are equal. To calculate the values ​​of the angles of a right-angled 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.е. angles α and β are 45º.

If the value of one of the acute angles is known, the second one can be found by 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 other two will be sharp. To find them, you need to know that:

other methods

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 the right angle to the hypotenuse. Let s be the median drawn from the right angle to the midpoint of the hypotenuse, h be the height. In this case it turns out that:

• sinα=b/(2*s); sinβ=a/(2*s).
• cosα=a/(2*s); cos β=b/(2*s).
• sinα=h/b; sinβ=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 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).

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

Quick article navigation

The length of the sides of a right triangle

The solution of 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 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 an angle, height, area of ​​a triangle, radius of a 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, transferring all known parameters to one side of the equal sign:

2) Substitute 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 the angle and the other two sides, the solution is reduced to calculating the trigonometric equation. Knowing the relationship of 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 is not enough.

1) First, we calculate sin γ by 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 sine theorem:

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.