Perimeter of Triangle

A triangle is the most basic polygon of geometry with three sides, angles, and vertices. The triangle can be differentiated based on its measure of sides and angles, with the sum of all the angles of the triangle always equal to 180°. The Perimeter is the basic concept of the triangle, which helps in understanding the size of the triangle and various practical and theoretical applications. 

1.0Introduction to Perimeter of a Triangle

The Perimeter of the triangle is the sum of all the lengths of a triangle, or in simple words, it is the total length of its boundary. To calculate the perimeter of a triangle, one simply needs to add up all the lengths of the three sides of the triangle. 

The Perimeter of a Triangle Formula 

Imagine a triangle with sides of lengths a, b, and c. The perimeter P of the triangle is calculated with the given formula: 

P = a + b + c 

Important Points to Remember 

• The unit to measure the perimeter of the triangle is always the same as the units of the sides of the triangle. For instance, if the sides are given in centimetres, then the perimeter will always be measured in centimetres only. 
• It is necessary to convert the unit of the lengths of the sides into only one unit to avoid any blunder during the calculation of perimeters. 
• The measurement of the perimeter of a triangle is always a positive value. 
• The perimeter of a triangle is only affected by the lengths and not the angles of a triangle. 
• Note that the above-mentioned formula is applicable to all types of triangles to find the perimeter of a triangle with slight changes for every type. 

2.0Types of Triangles

As mentioned above, the triangles are of different types, with some specific characteristics of each triangle. Although the formula for calculating the perimeter remains the same for all the triangles, the length of the side may vary. Hence, the formula for the perimeter of each triangle can also vary as follows: 

3.0Semi-perimeter of a Triangle

The semi-perimeter is half of the perimeter of any geometric figure and is denoted by “s”. The formula for the semi-perimeter of a triangle is given as follows: 

$s=\frac{a+b+c}{2}$

4.0The Perimeter of Triangle Questions 

Problem 1: Find the perimeter of triangle ABC, Given that ABC is an equilateral triangle with each side equal to 5 cm. Also, find the semi-perimeter of the triangle. 

Solution:  Let the side of triangle ABC = a = 5 cm.

The perimeter of the equilateral triangle P = 3a 

= $3\times 5=15cm$

The semi-perimeter of the triangle = P/2

= 15/2 

= 7.5cm 

Problem 2: A Farmer has a triangular field of maise with sides measuring 10 meters, 15 meters, and 20 meters. How much fencing material does the farmer need to enclose the garden? Also, find the cost of fencing at the rate of 20 Rs per meter.

Solution: According to the question, 

Length of Fencing material = Perimeter of the triangular field

Length of Fencing material P = 10 + 15 + 20 = 45m 

Cost of fencing the field = $Perimeterofthefield\times Rateoffencing$

Cost of fencing the field = $45\times 20=900Rs$

Problem 3: A ladder is leaning against a wall, forming a right-angled triangle. The distance from the base of the ladder to the wall is 6 meters, and the ladder reaches a height of 8 meters from the ground. What is the length of the ladder?

Solution: According to the question, 

Let the distance from the base of the ladder to the wall = a = 6m 

Let the height of the wall = b = 8m 

Let the length of the ladder = c 

By using Pythagoras theorem: 

a² + b² = c²

6² + 8² = c²

c = $\sqrt{36+64}=\sqrt{100}$

c=10m

Hence, the length of the ladder = 10m