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.)

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 the 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!!

ANDREY PROKIP: “MY LOVER IS RUSSIAN ECOLOGY. YOU NEED TO INVEST IN IT!”
On September 4-5, the environmental forum “Climatic Shape of Cities” was held. The initiator of the event is the C40 organization, which was founded in 2005 by the UN. The main task of the form and cities is to control climate change cities.
As practice has shown, in contrast to social events and “meetings in nightclubs,” there were few deputies and public figures. Among those who did identify concerns environmental situation was Prokip Adrey Zinovievich. He took an active part in all plenary sessions together with the Special Representative of the President Russian Federation on climate issues Ruslan Edelgeriev, Deputy Mayor of Moscow for Housing and Communal Services Pyotr Biryukov, as well as foreign representatives - the mayor of the Italian city of Savona - Ilario Caprioglio. Participants presented their projects and also discussed strategies to curb the rise in global temperatures, as well as proposed practical solutions sustainable development cities.
ANDREY PROKIP ABOUT SHASHLIKS, DEPUTIES AND GREEN BUILDING
Of particular interest to Russian side caused a presentation by speakers, among whom were European architects, scientists and the Mayor of Savona. The topic of the speech was the TOP direction - “green construction”. As Andrey Prokip himself stated, “it is important to correctly redistribute resources, as well as take into account European construction standards for a metropolis like Moscow. It is necessary for Russia to take a course towards “green financing” at the Federal level, especially since it is economically feasible and, as practice shows, profitable.” He also expressed concerns about the deterioration of the health of Russians due to environmental disasters and non-compliance with environmental standards for waste disposal by large and small industrial enterprises" He was also confirmed in his fears thanks to the speech of Francesco Zambona, a professor at the WHO European Office for Investment in Health.
With characteristic humor, Andrei addressed famous people who were invited to the forum, but never showed up, with a call to “remember nature, not only when they want barbecue or go fishing. After all, the health of the entire people depends on the benevolence of nature, which, unfortunately, includes them.”
In addition to passionate speeches about Andrei Zinovievich’s new “lover-nature” and the importance of taking responsibility for environment itself, a significant event of the forum was plenary session on the topic “How to raise a new generation.” The forum participants were unanimous in the opinion that it is necessary to educate not only children, but also the adult generation. It is very important to instill responsibility towards nature in everyday behavior, as well as in business.
A special project “learning to live in a civilized manner” will be launched for Moscow. This educational project for all segments of the population and age categories. But no matter how wonderful the theory and good intentions are, the saying “until the roast rooster pecks, the fool will not cross himself” is still relevant for Russia.
According to Timothy Netter, a famous theater director, art can change everything. In one of his speeches, he talked about how the idea of ​​preserving nature should be presented in theater and cinema and how important it is to educate people through art to be responsible for what will happen to us and nature tomorrow.
Students from Russian universities attracted the attention of Rentv operators and Andrey Prokirpa by presenting a project on environmentally friendly technology for the production of containers that are resistant to moisture and temperature. This is a very pressing problem, since laws are being passed around the world against plastic containers, which, by the way, take more than 30 years to decompose, pollute the soil and cause the death of animals.
It is encouraging that Moscow is one of 94 participating cities in the C40 organization and this is the third time the forum has been held, which every year attracts the attention of more and more famous personalities and citizens.

A right triangle is found in reality on almost every corner. Knowledge of the properties of a given figure, as well as the ability to calculate its area, will undoubtedly be useful to you not only for solving geometry problems, but also in life situations.

Triangle geometry

In elementary geometry, a right triangle is a figure that consists of three connected segments that form three angles (two acute and one straight). The right triangle is an original figure characterized by a number of important properties that form the foundation of trigonometry. Unlike a regular triangle, the sides of a rectangular figure have their own names:

• The hypotenuse is the longest side of a triangle, opposite the right angle.
• Legs are segments that form a right angle. Depending on the angle under consideration, the leg can be adjacent to it (forming this angle with the hypotenuse) or opposite (lying opposite the angle). There are no legs for non-right triangles.

It is the ratio of the legs and hypotenuse that forms the basis of trigonometry: sines, tangents and secants are defined as the ratio of the sides of a right triangle.

Right triangle in reality

This figure has become widespread in reality. Triangles are used in design and technology, so calculating the area of ​​a figure has to be done by engineers, architects and designers. The bases of tetrahedrons or prisms - three-dimensional figures that are easy to meet in everyday life - have the shape of a triangle. Additionally, a square is the simplest representation of a "flat" right triangle in reality. A square is a metalworking, drawing, construction and carpentry tool that is used to construct angles by both schoolchildren and engineers.

Area of ​​a triangle

Square geometric figure is a quantitative assessment of how much of the plane is bounded by the sides of the triangle. The area of ​​an ordinary triangle can be found in five ways, using Heron's formula or using such variables as the base, side, angle and radius of the inscribed or circumscribed circle. The most simple formula area is expressed as:

where a is the side of the triangle, h is its height.

The formula for calculating the area of ​​a right triangle is even simpler:

where a and b are legs.

Working with our online calculator, you can calculate the area of ​​a triangle using three pairs of parameters:

• two legs;
• leg and adjacent angle;
• leg and opposite angle.

In problems or everyday situations you will be given different combinations of variables, so this form of the calculator allows you to calculate the area of ​​a triangle in several ways. Let's look at a couple of examples.

Real life examples

Ceramic tile

Let's say you want to cover the kitchen walls with ceramic tiles, which have the shape of a right triangle. In order to determine the consumption of tiles, you must find out the area of ​​one cladding element and the total area of ​​the surface being treated. Suppose you need to process 7 square meters. The length of the legs of one element is 19 cm, then the area of ​​the tile will be equal to:

This means that the area of ​​one element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that to finish 7 square meters of wall you will need 7/0.01805 = 387 elements of facing tiles.

School task

Let's say in a school geometry problem you need to find the area of ​​a right triangle, knowing only that the side of one leg is 5 cm, and the opposite angle is 30 degrees. Our online calculator comes with an illustration showing the sides and angles of a right triangle. If side a = 5 cm, then its opposite angle is angle 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 ​​a given triangle, but also determines the length of the adjacent leg and hypotenuse, as well as the value of the second angle.

Conclusion

Right triangles are found in our lives literally on every corner. Determining the area of ​​such figures will be useful to you not only when solving school assignments in geometry, but also in everyday and professional activities.

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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