Perimeter of a triangle: concept, characteristic, methods of determination

The triangle is one of the fundamental geometric figures, representing three intersecting segments of straight lines. This figure was known to scholars of Ancient Egypt, Ancient Greece and Ancient China, who derived most of the formulas and laws used by scientists, engineers and designers to this day.

The main components of the triangle are:

• Vertices are the intersection points of segments.

• The sides are the intersecting segments of lines.

Proceeding from these components, formulate such concepts as perimeter of a triangle, its area, inscribed and circumscribed circle. From school it is known that the perimeter of a triangle is a numerical expression of the sum of all its three sides. At the same time, a great number of formulas are known for finding a given value, 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 in the case when the numerical values of all its three sides (x, y, z) are known, as a consequence:

P = x + y + z

2. The perimeter of an equilateral triangle can be found if we recall that in this figure all sides, however, like all angles, 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, unlike an equilateral triangle, only two lateral sides have the same numerical value, so in this case the perimeter in general will be as follows:

P = 2x + y

4. The following methods are necessary in cases where the numerical values of not all sides 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 defining the third side and the known angle. In this case, this third party will be found by the formula:

Z = 2x + 2y-2xycosβ

Proceeding from 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 not 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 a circle inscribed in it are used to find the perimeter of a triangle. This formula is also known to most people since the school day:

P = 2S / r (S is the area of the circle, while r is its radius).

From all of the above, it can be seen that the perimeter of a triangle can be found in a variety of ways, based on the data that the researcher owns. In addition, there are several more particular cases of finding a given value. Thus, the perimeter is one of the most important quantities and characteristics of a right-angled triangle.

As is known, such a triangle is a figure whose two sides form a right angle. The perimeter of a right-angled triangle is found through the numerical expression of the sum of both the legs and the hypotenuse. If the researcher knows only two sides, the remaining one can be calculated using the famous Pythagoras theorem: z = (x2 + y2) if both legs are known, or x = (z2 - y2) if the hypotenuse and the cat are known.

In the event that the length of the hypotenuse and one of the angles adjacent to it are known, the other two sides are given by the formulas: x = z sinβ, y = z cosβ. In this case, the perimeter of a right-angled triangle will be:

P = z (cosβ + sinβ + 1)

Also a special case is the calculation of the perimeter of a regular (or equilateral) triangle, that is, a figure in which all sides and all angles are equal. Calculation of the perimeter of such a triangle on the known side does not constitute any 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 must be memorized in order to be successfully applied in practice.