How to Find the Perimeter of a Triangle

Introduction:-

Triangles are one of the fundamental shapes in geometry. Understanding their properties and being able to calculate their various attributes is essential for both students and professionals and teachers in various fields. One of the most basic calculations for a triangle is finding its perimeter, which is the sum of the lengths of all its sides. In this article, I will try to explain into the various methods of How To Find The Perimeter Of A Triangle, including different types of triangles and the different cases to consider.

Explanation:-

Understanding the Basics
Before I start this topic,How To Find The Perimeter Of A Triangle, let’s establish a few key concepts and definitions:

Triangle Types:

Based on sides triangles are divided in three categories:
(a)Equilateral Triangle: A triangle whose all three sides are equal is called equilateral triangle.

In the picture,
ABC is an equilateral triangle,whose
AB = BC = AC
(b)Isosceles Triangle: A triangle whose any two sides are equal and third one is different is called isosceles triangle.

In the picture,
PQR is an isosceles triangle whose
PQ = PR
©Scalene Triangle: A triangle whose all three sides are different to one another is called a scalene triangle.

In the picture,
XYZ is a scalene triangle whose,
XY ≠ YZ ≠ ZX

Sides and Vertices:

(i)The sides of a triangle are the line segments that form the boundaries of the triangle.
(ii)The vertices of a triangle are the points where the sides meet.

Perimeter:

The perimeter of a triangle is the total length or sum of all its sides.

Sides of a Triangle:

In an equilateral triangle, all sides have the same length, which we’ll denote as ‘a.’
In an isosceles triangle, two sides have the same length, and the third side is ‘b.’
In a scalene triangle, all three sides have different lengths, which we’ll denote as ‘a,’ ‘b,’ and ‘c.’
With these basic concepts in mind, let’s discuss How To Find The Perimeter Of A Triangle in various scenarios

Procedure 1:

Perimeter of an Equilateral Triangle:
An equilateral triangle is a special case where all sides are of equal length. To find its perimeter, we will simply multiply the length of one side (‘a’) by 3, as there are three equal sides.
So,Perimeter of an Equilateral Triangle (P) = 3a

Some problems related on equilateral triangle:-

(a) The measure of each side of an equilateral triangle is 5 cm, Find its perimeter.

Solution:-

∵ Each side of the equilateral triangle(a) = 5cm
∴ Perimeter of the equilateral triangle = 3a
= (3 × 5)cm
= 15 cm
(b) The perimeter of an equilateral triangle is 21cm, find its each side.

Solution:-

∵The perimeter of equilateral triangle = 21 cm
∴ Its each side measure = 21/3
= 7cm

Procedure 2:

Perimeter of an Isosceles Triangle:

An isosceles triangle has two sides of equal length (‘a’) and one side of different length (‘b’). To calculate its perimeter, we will add the lengths of the two equal sides (‘a’) and the different side (‘b’).
So,
Perimeter of an Isosceles Triangle
(P) = 2a + b

Some problems related on isosceles triangle:

(a) The measure of two equal sides of an isosceles triangle is 7m each and third side is 5m, find the perimeter of the isosceles triangle.

Solution:-

∵ The measure of each equal side of the isosceles triangle(a) = 7m
The measure of third side (b) = 5m
∴ Perimeter of the isosceles triangle = 2a + b
= (2 × 7 + 5)m
= (14 + 5)m
= 19 m

(b)The perimeter of an isosceles triangle is 27m,the measure of its third side is 3m, find the measure of its each equal side.

Solution:- 1st Method:

∵ Perimeter of isosceles triangle = 27m
Measure of its third side(b) = 3m
Let, measure of each equal side = a meter
A/Q,
2a + b = 27
=> 2 × a + 3 = 27
=> 2 × a = 27–3
=> 2 × a = 24
=> a = 24/2
=> a = 12
∴ Measure of each equal side of the isosceles triangle (a) = 12 m

2nd Method:

∵ Perimeter of isosceles triangle = 27m
Measure of its third side(b) = 3m
∴Measure of each equal side(a) = (Perimeter — b) / 2
= (27 -3)/2
= 24/2
= 12 m

©The perimeter of an isosceles triangle is 20 cm, its equal sides measure 4cm, Find its third side.

Solution:- 1st Method:

The perimeter of an isosceles triangle = 20cm
Measure of its equal sides (a) = 4cm
Let, measure of its third side = b cm
A/Q,
2a + b = 20
=> 2 × 4 + b = 20
=> 8 + b = 20
=> b = 20–8
=> b = 12
∴ Measure of third side(b) = 12 cm

2nd Method:

The perimeter of an isosceles triangle = 20cm
Measure of its equal sides (a) = 4cm
∴ Measure of its third side (b) = perimeter — 2a
= (20–2 × 4)
= 20–8
= 12 cm

Procedure 3:

Perimeter of a Scalene Triangle:

In a scalene triangle, all three sides have different lengths (‘a,’ ‘b,’ and ‘c’). To find the perimeter, we will simply add the lengths of all three sides.
∴Perimeter of a Scalene Triangle
(P) = a + b + c

Some Problems on scalene triangle:

(a) Three sides of a scalene triangle are 3m, 4m, and 5m respectively, Find the perimeter of the triangle.

Solution:-

∵ 1st side (a) = 3m
2nd side (b) = 4m
and 3rd side © =5m
∴ Perimeter of scalene triangle (P) = (3 + 4 + 5) = 12 m

A/Q,
a + b + c = P
=> 6 + 7 + c = 15
=> 13 + c = 15
=> c = 15–13
=> c = 2
∴ Third side © = 2 cm

(c)The perimeter of a scalene triangle is 30m and sum of its two side is 20m,find the third side.

Solution:-

∵ Perimeter of scalene triangle = 30m
Sum of its two sides = 20m
∴ Measure of third side = Perimeter — Sum of its two sides
= (30–20)m
= 10m

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