Perimeter of a triangle - formula and explanation

Perimeter of a triangle - formula and explanation

The perimeter of a triangle is the total length of all its sides. The easiest way to find it is to add up the lengths of all its sides, but if you do not know the length of at least one side of the triangle, you must first find it. In today’s article, we shall analyze the perimeter of a triangle, its formula and explanation.

Perimeter of a triangle

Image: pixabay.com
Source: UGC

Want to know how to find the perimeter of a triangle? Read on.

Perimeter of an isosceles triangle

A perimeter is the sum of all sides of the shape. This characteristic, along with the area, is equally in demand for all figures. The formula of the perimeter of an isosceles triangle logically follows from its properties, but the formula is not as complicated as the acquisition and consolidation of practical skills.

Perimeter of a triangle formula

Two sides of an isosceles triangle are equal. This results from the definition and can be clearly seen even from the name of the figure. It is from this property that the perimeter formula results:

P = 2a + b, where b is the base of the triangle, a is the value of the side.

Perimeter of a triangle - formula and explanation

Image: obrazovaka.ru
Source: UGC

Fig. 1. Isosceles triangle

From the formula it is clear that to find the perimeter, it is enough to know the size of the base and one of the sides. Consider several tasks for finding the perimeter of an isosceles triangle. We will solve exercises as the complexity increases, this will allow us to better understand the way of thinking that must be followed to find the perimeter.

Task 1

In an isosceles triangle, the base is 6, and the height drawn to this base is 4. It is necessary to find the perimeter of the figure.

Perimeter of a triangle - formula and explanation

Image: obrazovaka.ru
Source: UGC

Fig. 2. Drawing to task 1

The height of an isosceles triangle drawn to the base is also the median and height. This property is very often used in solving problems related to isosceles triangles.

The ABC triangle with height BM is divided into two right triangles: ABM and BCM. In the triangle ABM, the leg BM is known, the leg AM is half the base of the triangle ABC, since VM is the median of the bisector and height. By the Pythagorean theorem, we find the value of the hypotenuse AB.

Perimeter

Image: obrazovaka.ru
Source: UGC

READ ALSO: Who is the father of mathematics? - Meet 3 most prominent ancient scientists

Find the perimeter: P = AC + AB * 2 = 6 + 5 * 2 = 16

Task 2

In an isosceles triangle, the height drawn to the base is 10, and the acute angle at the base is 30 degrees. need to find the perimeter of the triangle.

Perimeter of a triangle - formula and explanation

Fig. 3. Drawing to task 2

This task is complicated by the lack of information about the sides of the triangle, but knowing the height and angle values, you can find the leg AH in the right-angled triangle ABH, and then the solution follows the same scenario as in task 1.

Find AH through the sine value:

Formula 2

Image: obrazovaka.ru
Source: UGC

The sine of 30 degrees is a tabular value.

Express the desired side:

Formula 3

Image: obrazovaka.ru
Source: UGC

Through cotangent find the value of AH:

Formula 4

Image: obrazovaka.ru
Source: UGC

the resulting value is rounded to the hundredth.

Find the base:

AC = AH * 2 = 17.32 * 2 = 34.64

Now that all the required values are found, we define the perimeter:

P = AC + 2 * AB = 34.64 + 2 * 20 = 74.64

Task 3

In the isosceles triangle ABC, the area is known, which is equal to

Formula 5

and an acute angle at the base of 30 degrees. Find the perimeter of the triangle.

The values in the condition are often given as the product of the root by the number. This is done in order to maximally protect the subsequent decision from errors. It is better to round the result at the end of calculations.

With such a formulation of the problem, it may seem that there are no solutions, because it is difficult to express one of the parties or the height of the available data. Let's try to solve it differently.

Denote the height and half of the base in Latin letters: BH = h and AH = a

Then the base will be: AC = AH + HC = AH * 2 = 2a

Square:

Formula 5

Image: obrazovaka.ru
Source: UGC

On the other hand, the value of h can be expressed from the triangle ABH through the tangent of the acute angle. Why precisely tangent? Because in the triangle ABH we have already identified two legs a and h. It is necessary to express one through the other. Two legs together tangent and cotangent. Traditionally, the cotangent and cosine are addressed only if the tangent or sine do not fit. This is not a rule, you can solve it as it is convenient, just so accepted.

Formula 6

Image: obrazovaka.ru
Source: UGC

the resulting value is rounded to the hundredth.

Through the Pythagorean theorem we find the side of the triangle:

Formula 7

Image: obrazovaka.ru
Source: UGC

Substitute the values in the perimeter formula:

P = AB * 2 + AH * 2 = 4.62 * 2 + 4 * 2 = 17.24

What have we learned today?

We understood in detail all the intricacies of finding the perimeter of a triangle. We solved three problems of different levels of complexity, showing by example how typical problems for solving a triangle perimeter.

READ ALSO: Best mathematician in the world

Source: Legit.ng

Mailfire view pixel