A triangle is one of the fundamental geometric figures, which are three intersecting line segments. This figure was already known to scientists of Ancient Egypt, Ancient Greece and Ancient China, who have deduced most of the formulas and laws used by scientists, engineers and designers so far.
The main components of the triangle include:
• Vertices - intersection points of segments.
• Parties - intersecting line segments.
Based on these components, concepts such as the perimeter of a triangle, its area, inscribed and described circle are formulated. It has been known since school that the perimeter of a triangle is a numerical expression of the sum of all three of its sides. At the same time, there are a great many formulas for finding this quantity, depending on the initial data that the researcher has in one or another case.
1. The simplest way to find the perimeter of a triangle is used when the numerical values of all three of its sides (x, y, z) are known, as a result:
P = x + y + z
2. The perimeter of an equilateral triangle can be found if we recall that this figure has all sides, however, like all angles, they are equal. Knowing the length of this side, the perimeter of an equilateral triangle can be determined by the formula:
P = 3x
3. In an isosceles triangle, in contrast to an equilateral triangle, only two sides have the same numerical value, therefore, in this case, in general terms, the perimeter will be as follows:
P = 2x + y
4. The following methods are necessary in cases where the numerical values of not all parties are known. For example, if the study has data on two sides, and the angle between them is known, then the perimeter of the triangle can be found by determining the third side and the known angle. In this case, this third party will be found by the formula:
z = 2x + 2y-2xycosβ
Based on this, the perimeter of the triangle will be equal to:
P = x + y + 2x + (2y-2xycos β)
5. In the case when the length of no more than one side of the triangle is initially given and the numerical values of the two angles adjacent to it are known, the perimeter of the triangle can be calculated based on the sine theorem:
P = x + sinβ x / (sin (180 ° -β)) + sinγ x / (sin (180 ° -γ))
6. There are cases when the known parameters of the circle inscribed in it are used to find the perimeter of a triangle. This formula is also known to most since school days:
P = 2S / r (S is the area of the circle, while r is its radius).
From the foregoing, it can be seen that the perimeter of the triangle can be found in many ways, based on the data that the researcher owns. In addition, there are several more particular cases of finding this value. So, the perimeter is one of the most important values and characteristics of a right triangle.
As you know, such a triangle is a figure whose two sides form a right angle. The perimeter of a right triangle is found through a numerical expression of the sum of both legs and hypotenuse. If the researcher knows only two sides, the rest can be calculated using the famous Pythagorean theorem: z = (x2 + y2) if both legs are known, or x = (z2 - y2) if the hypotenuse and leg are known.
In the event that the length of the hypotenuse and one of the corners adjacent to it are known, then the other two sides are found by the formulas: x = z sinβ, y = z cosβ. In this case, the perimeter of a right triangle will be equal to:
P = z (cosβ + sinβ +1)
Another special case is the calculation of the perimeter of a regular (or equilateral) triangle, that is, such a figure in which all sides and all angles are equal. Calculation of the perimeter of such a triangle on the known side is no problem, however, often the researcher knows some other data. So, if the radius of the inscribed circle is known, the perimeter of the regular triangle is found by the formula:
P = 6√3r
And if the radius of the circumscribed circle is given, the perimeter of the regular triangle will be found as follows:
P = 3√3R
Formulas need to be remembered in order to be successfully applied in practice.