Ways to find an angle in a right triangle - calculation formulas. Online calculator. Solving triangles Calculation of angles and lengths in a right triangle

A triangle is a geometric number consisting of three segments that connect three points that do not lie on the same line. The points that form a triangle are called its points, and the segments are side by side.

Depending on the type of triangle (rectangular, monochrome, etc.), you can calculate the side of the triangle in different ways, depending on the input data and the conditions of the problem.

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To calculate the sides of a right triangle, the Pythagorean theorem is used, according to which the square of the hypotenuse equal to the sum square feet.

If we label the legs as "a" and "b" and the hypotenuse as "c", then the 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 angle opposite the base, "a" is the adjacent angle to calculate the pages can use the following formulas:

Two corners and a 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 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 information known about it.

You'll need it

• Depending on the case, skills 2 three sides of the triangle, as well as one of its acute angles.

instructions

first Method 1. If all three pages are known triangle Then, regardless of whether 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 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 this way: P = (1 + sin?

fourth Method 4. They say that in the right triangle the length of one leg is equal to a and, on the contrary, has an acute angle. Then calculate perimeter This triangle will be carried out according to the formula: P = a * (1 / tg?

1/son? + 1)

fifths 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 basis of all mathematics. Determines the relationship between the sides of a true triangle. There are now 367 proofs of this theorem.

instructions

first The classic school formulation of the Pythagorean theorem sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs.

To find the hypotenuse in right triangle two Catets, you must contact to build a square of 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 the hypotenuse, which is 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 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 unfounded number.

third If the triangles are legs 3 and 4, then hypotenuse = 25 = 5. When you take the square root, you get natural number. The numbers 3, 4, 5 form a Pygagorean triplet, 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, 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 don't need A.

fifths The Pythagorean theorem is a special case, greater than the general cosine theorem, which establishes the 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 the side in a right triangle that is opposite the 90 degree angle.

instructions

first In the case of known catheters, as well as the 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 = C1 (or C2) / sin, H = C1 (or C2?) / 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 a rectangle triangle. It is located at a right angle. Method for finding the hypotenuse of a rectangle triangle depending on the source data.

instructions

first If your legs are perpendicular triangle, then the length of the hypotenuse of the rectangle triangle can be discovered by a 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 one of the legs is known and at an acute angle, the formula for finding the hypotenuse will depend on the presence or absence of under certain angle in relation to the known leg - adjacent (the leg is located close), or vice versa (the opposite case is located nego.V of the specified angle is equal to the fraction of the hypotenuse of the leg in the cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio sinusoidal angles: da = a / sin.

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Useful tips
An angular triangle whose sides are related 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 Jero's triangles, in which pages and area are represented by integers.

A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other is called the legs.

If you want to find how a right triangle is formed by some properties of regular triangles, namely the fact that the sum of the acute angles is 90°, which is used, and the fact that the length of the opposite leg is half the hypotenuse is 30°.

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Cropped triangle

One of the properties of an equal triangle is that its two angles are equal.

To calculate the angle of a right congruent triangle, you need to know that:

• This is 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 equal to 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 often 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 remain sharp.

Calculate triangle online

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

other methods

The values ​​of the acute angles of a right triangle can be calculated from the average - 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 let 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 triangle or on both sides, then trigonometric identities are used to determine the values ​​of the 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. General formula to find triangular triangle:

where P is the circumference of the triangle, a, b and c of 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 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 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 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 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 ​​a parallelogram is the same as the product of its base height, the area of ​​the triangle will be equal to half of this product. Thus, 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 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 triangle is equal to the product 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 the length, and the height is reduced to the 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.

Triangle Definition

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

Online calculator

There are triangles various types. For example, there is an equilateral triangle (one in which all sides are equal), isosceles (two sides are equal in it) and a right triangle (in which one of the angles is straight, i.e., equal to 90 degrees).

The area of ​​a triangle can be found different ways depending on what elements of the figure are known from the conditions of the problem, be it angles, lengths, or even the radii of circles associated with the triangle. Let's look at each method separately with examples.

Formula for the area of ​​a triangle based on its base and height

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

A a a- 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 this into the formula for area and 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. sq.)

Formula for the area of ​​a triangle based on 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- 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 (cm), 4 (cm), 5 (cm).

Solution

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

Let's find half 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 ​​the 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 square)

Formula for the area of ​​a triangle given 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- length of the side of the triangle;
β , γ \beta, \gamma β , γ - angles adjacent to the side a a a.

Example

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

Formula for the area of ​​a triangle based on three sides and the radius of the circumcircle

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

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

Example

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

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 (cm2)

Formula for the area of ​​a triangle based on three sides and the radius of the inscribed circle

S = p ⋅ r S=p\cdot rS= p⋅ r,

p pp- half 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- the radius of the circle inscribed in the triangle.

Example

Let the radius of the inscribed circle be 2 (cm). We will 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 (cm. sq.)

Formula for the area of ​​a triangle based 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 - angle between sides b bb And c cc.

Example

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

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).

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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 comes down 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 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.

In mathematics, when considering a triangle, a lot of attention is paid to its sides. Because these elements form this geometric figure. The sides of a triangle are used to solve many geometry problems.

Definition of the concept

Segments connecting three points that do not lie on the same line are called sides of a triangle. The elements under consideration limit part of the plane, which is called the interior of this geometric figure.

Mathematicians in their calculations allow generalizations regarding the sides of geometric figures. Thus, in a degenerate triangle, three of its segments lie on one straight line.

Characteristics of the concept

Calculating the sides of a triangle involves determining all other parameters of the figure. Knowing the length of each of these segments, you can easily calculate the perimeter, area and even the angles of the triangle.

Rice. 1. Arbitrary triangle.

By summing the sides of a given figure, you can determine the perimeter.

P=a+b+c, where a, b, c are the sides of the triangle

And to find the area of ​​a triangle, then you should use Heron's formula.

$$S=\sqrt(p(p-a)(p-b)(p-c))$$

Where p is the semi-perimeter.

The angles of a given geometric figure are calculated using the cosine theorem.

$$cos α=((b^2+c^2-a^2)\over(2bc))$$

Meaning

Some properties of this geometric figure are expressed through the ratio of the sides of a triangle:

• Opposite the smallest side of a triangle is its smallest angle.
• The external angle of the geometric figure in question is obtained by extending one of the sides.
• Against equal angles a triangle has equal sides.
• In any triangle, one of the sides is always greater than the difference of the other two segments. And the sum of any two sides of this figure is greater than the third.

One of the signs that two triangles are equal is the ratio of the sum of all sides of the geometric figure. If these values ​​are the same, then the triangles will be equal.

Some properties of a triangle depend on its type. Therefore, you should first take into account the size of the sides or angles of this figure.

Forming triangles

If the two sides of the geometric figure in question are the same, then this triangle is called isosceles.

Rice. 2. Isosceles triangle.

When all the segments in a triangle are equal, you get an equilateral triangle.

Rice. 3. Equilateral triangle.

It is more convenient to carry out any calculation in cases where an arbitrary triangle can be classified as a specific type. Because then finding the required parameter of this geometric figure will be significantly simplified.

Although a correctly chosen trigonometric equation allows you to solve many problems in which an arbitrary triangle is considered.

What have we learned?

Three segments that are connected by points and do not belong to the same straight line form a triangle. These sides form a geometric plane, which is used to determine the area. Using these segments, you can find many important characteristics of a figure, such as perimeter and angles. The aspect ratio of a triangle helps to find its type. Some properties of a given geometric figure can only be used if the dimensions of each of its sides are known.

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

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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 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 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).