How to find the perimeter of a triangle?

How to find the perimeter of a triangle? Each of us asked this question while studying at school. Let's try to remember everything that we know about this amazing figure, and also answer the question asked.

The answer to the question of how to find the perimeter of a triangle is usually quite simple - you only need to perform the procedure of adding the lengths of all its sides. However, there are a few simple methods of the desired magnitude.

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In the event that the radius (r) of the circle that is inscribed in the triangle and its area (S) are known, then it is quite easy to answer the question of how to find the perimeter of a triangle. To do this, you need to use the usual formula:

P = 2S / r

If two angles are known, say, α and β, which are adjacent to the side, and the length of the side itself, then the perimeter can be found with the help of a very popular formula, which has the form:

Sinβ ∙ a / (sin (180 ° - β - α)) + sinα ∙ a / (sin (180 ° - β - α)) + a

If you know the lengths of adjacent sides and the angle β that lies between them, then in order to find the perimeter, you need to use the cosine theorem. Perimeter is calculated by the formula:

P = b + a + √ (b2 + a2 - 2 ∙ b ∙ a ∙ cosβ),

Where b2 and a2 are squares of lengths of adjacent sides. The radicand is the length of the third side, which is unknown, expressed by means of the cosine theorem.

If you do not know how to find the perimeter of an isosceles triangle, then, in fact, there is nothing complicated. Calculate it using the formula:

P = b + 2a,

Where b is the base of the triangle, and a is its lateral sides.

To find the perimeter of a regular triangle, one should use the simplest formula:

P = 3a,

Where a is the side length.

How to find the perimeter of a triangle if only the radii of circles are known that are described near or inscribed in it? If the triangle is equilateral, then the formula should be applied:

P = 3R√3 = 6r√3,

Where R and r are the radii of the circumscribed and inscribed circle, respectively.

If the triangle is isosceles, then the formula is applicable to it:

P = 2R (sinβ + 2sinα),

Where α is the angle that lies at the base, and β is the angle that opposes the base.

Often, for solving mathematical problems, it requires deep analysis and a specific ability to find and output the required formulas, and this, as we all know, is quite a difficult job. Although some problems can be solved only with the help of a single formula.

Let's look at formulas that are basic to answering the question of how to find the perimeter of a triangle, with respect to the most diverse types of triangles.

Of course, the main rule for finding the perimeter of a triangle is this statement: to find the perimeter of a triangle, it is required to add the lengths of all its sides according to the corresponding formula:

P = b + a + c,

Where b, a and c are the lengths of the sides of the triangle, and P is the perimeter of the triangle.

There are several special cases of this formula. Let's say your task is formulated as follows: "How to find the perimeter of a right-angled triangle?" In this case, you should use the following formula:

P = b + a + √ (b2 + a2)

In this formula, b and a are the direct lengths of the legs of a right triangle. It is easy to guess that instead of the side with (hypotenuse), the expression obtained by the theorem of the great scholar of antiquity - Pythagoras is used.

If you want to solve a problem where triangles are similar, then it would be logical to use this statement: the perimeter ratio corresponds to the similarity coefficient. Let's say you have two such triangles - ΔABC and ΔA1B1C1. Then, to find the similarity coefficient, it is necessary to divide the perimeter ΔABC by the perimeter ΔA1B1C1.

In conclusion, it can be noted that the perimeter of the triangle can be found using a variety of techniques, depending on the source data that you have. It must be added that there are some particular cases for right-angled triangles.