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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 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.
Opposite equal angles of a triangle are 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

Values of acute angles 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 drawn 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).

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

The area of a geometric figure is a quantitative estimate 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 simplest formula for 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 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.

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.
The students attracted the attention of Rentv operators and Andrey Prokirpa Russian universities, 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.

In geometry, an angle is a figure that is formed by two rays that emerge from one point (called the vertex of the angle). In most cases, the unit of measurement for angle is degree (°) - remember that a full angle, or one revolution, is 360°. You can find the angle value of a polygon by its type and the values of other angles, and if given a right triangle, the angle can be calculated from two sides. Moreover, the angle can be measured using a protractor or calculated using a graphing calculator.

Steps

How to find interior angles of a polygon

Count the number of sides of the polygon. To calculate the interior angles of a polygon, you first need to determine how many sides the polygon has. Note that the number of sides of a polygon is equal to the number of its angles.

For example, a triangle has 3 sides and 3 interior angles, and a square has 4 sides and 4 interior angles.

Calculate the sum of all interior angles of the polygon. To do this, use the following formula: (n - 2) x 180. In this formula, n is the number of sides of the polygon. The following are the sums of the angles of commonly encountered polygons:
- The sum of the angles of a triangle (a polygon with 3 sides) is 180°.
- The sum of the angles of a quadrilateral (a polygon with 4 sides) is 360°.
- The sum of the angles of a pentagon (a polygon with 5 sides) is 540°.
- The sum of the angles of a hexagon (a polygon with 6 sides) is 720°.
- The sum of the angles of an octagon (a polygon with 8 sides) is 1080°.
Divide the sum of all the angles of a regular polygon by the number of angles. A regular polygon is a polygon with equal sides and equal angles. For example, each angle of an equilateral triangle is calculated as follows: 180 ÷ 3 = 60°, and each angle of a square is calculated as follows: 360 ÷ 4 = 90°.
- An equilateral triangle and a square are regular polygons. And at the Pentagon building (Washington, USA) and road sign"Stop" shape of a regular octagon.
Subtract the sum of all known angles from the total sum of the angles of the irregular polygon. If the sides of a polygon are not equal to each other, and its angles are also not equal to each other, first add up the known angles of the polygon. Now subtract the resulting value from the sum of all the angles of the polygon - this way you will find the unknown angle.
- For example, if given that the 4 angles of a pentagon are 80°, 100°, 120° and 140°, add up these numbers: 80 + 100 + 120 + 140 = 440. Now subtract this value from the sum of all the angles of the pentagon; this sum is equal to 540°: 540 - 440 = 100°. Thus, the unknown angle is 100°.
Advice: the unknown angle of some polygons can be calculated if you know the properties of the figure. For example, in an isosceles triangle, two sides are equal and two angles are equal; In a parallelogram (which is a quadrilateral), opposite sides are equal and opposite angles are equal.

Measure the length of the two sides of the triangle. The longest side of a right triangle is called the hypotenuse. The adjacent side is the side that is near the unknown angle. The opposite side is the side that is opposite the unknown angle. Measure the two sides to calculate the unknown angles of the triangle.

Advice: use a graphing calculator to solve the equations, or find an online table with the values of sines, cosines, and tangents.

Calculate the sine of an angle if you know the opposite side and the hypotenuse. To do this, plug the values into the equation: sin(x) = opposite side ÷ hypotenuse. For example, the opposite side is 5 cm and the hypotenuse is 10 cm. Divide 5/10 = 0.5. Thus, sin(x) = 0.5, that is, x = sin -1 (0.5).