Online calculator.
Solving triangles.

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

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

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

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

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

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

Rules for entering numbers

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

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

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A little theory.

Theorem of sines

Theorem

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

Cosine theorem

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

Solving triangles

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

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

Solving a triangle using two sides and the angle between them

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

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

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

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

Solving a triangle by side and adjacent angles

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

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

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

Solving a triangle using three sides

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

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

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

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

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

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

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

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

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

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

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

Quick navigation through the article

Length of the sides of a right triangle

The solution to the problem follows from the theorem of the great mathematician Pythagoras. It says that the sum of the squares of the legs right triangle 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.

Triangle Definition

Triangle- This geometric figure, which is formed as a result of the intersection of three segments whose ends 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.)

Enter known triangle data

Side a

Side b

Side c

Angle A in degrees

Angle B in degrees

Angle C in degrees

Median on side a

Median to side b

Median on side c

Height on side a

Height on side b

Height on side c

Coordinates of vertex A

X Y

Vertex B coordinates

X Y

Coordinates of vertex C

X Y

Area of ​​triangle S

Semi-perimeter of the sides of a triangle p

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

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

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

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

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

Create a request and get an accurate answer.

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

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

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

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

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

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

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

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

Syntax

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

For site users, everything is done on this page.

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

the parameter is written as parameter=value

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

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

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

Side a

Side b

Side c

Semi-perimeter p

Angle A

Angle B

Angle C

Area of ​​triangle S

Height ha on side a

Height hb on side b

Height hc on side c

Median ma to side a

Median mb to side b

Median mc to side c

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

Examples

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

Triangle parameters according to given parameters

Side a = 8

Side b = 2.1283555449519

Side c = 7.5420719851515

Semi-perimeter p = 8.8352137650517

Angle A = 2.1882518638666 in degrees 125.37759631119

Angle B = 2.873202966917 in degrees 164.62240368881

Angle C = 1.221730476396 in 70 degrees

Area of ​​the triangle S = 8

Height ha on side a = 2

Height hb on side b = 7.5175409662872

Height hc on side c = 2.1214329472723

Median ma per side a = 3.8348889915443

Median mb per side b = 7.7012304590352

Median mc per side c = 4.4770789813853

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

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

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

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

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

and we get

Triangle parameters according to given parameters

Side a = 17

Side b = 11.401754250991

Side c = 13.453624047073

Semi-perimeter p = 20.927689149032

Angle A = 1.4990243938603 in degrees 85.887771155351

Angle B = 0.73281510178655 in degrees 41.987212495819

Angle C = 0.90975315794426 in degrees 52.125016348905

Area of ​​the triangle S = 76.5

Height ha on side a = 9

Height hb on side b = 13.418987695398

Height hc on side c = 11.372400437582

Median ma per side a = 9.1241437954466

Median mb per side b = 14.230249470757

Median mc per side c = 12.816005617976

Happy calculations!!