How to find the perimeter of a triangle? Each of us asked this question while studying at school. Let's try to recall everything we know about this amazing figure, as well as answer the question.
The answer to the question of how to find the perimeter of a triangle is usually quite simple - you just need to perform the procedure of adding the lengths of all its sides. However, there are a few more simple methods of the desired size.
Tips
In that case, if the radius (r) of the circle inscribed in the triangle and its area (S) are known, then it is quite simple to answer the question of how to find the perimeter of the triangle. To do this, you need to use the usual formula:
P = 2S / r
If two angles are known, for example, α and β, which are adjacent to the side, and the length of the side itself, then the perimeter can be found using a very, very popular formula, which has the form:
sinβ ∙ / (sin (180 ° - β - α)) + sinα ∙ / (sin (180 ° - β - α)) + a
If you know the lengths of adjacent sides and the angle β between them, then in order to find the perimeter, you need to use the cosine theorem. The perimeter is calculated by the formula:
P = b + a + √ (b2 + a2 - 2 ∙ b ∙ a ∙ cosβ),
where b2 and a2 are squared lengths of adjacent sides. The radical expression is the length of a third side that is unknown, expressed by the cosine theorem.
If you do not know how to find the perimeter of an isosceles triangle, then here, in fact, there is nothing complicated. Calculate it using the formula:
P = b + 2a,
where b is the base of the triangle, and a are its lateral sides.
To find the perimeter of a regular triangle, use the simplest formula:
P = 3a
where a is the length of the side.
How to find the perimeter of a triangle if only the radii of the circles that are described near it or are inscribed in it are known? 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 circles, 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 is opposite the base.
Often, solving mathematical problems requires in-depth analysis and a specific ability to find and derive the required formulas, and this, as many people know, is a rather difficult job. Although some problems can be solved with just one single formula.
Let's look at the formulas that are basic for answering the question of how to find the perimeter of a triangle, in relation 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, 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. Suppose your task is formulated as follows: "how to find the perimeter of a right 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 scientist 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 ratio of the perimeters corresponds to the similarity coefficient. Suppose you have two similar 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 special cases for right-angled triangles.