Perimeter of Triangle


The perimeter of a closed geometric object is the measurement of the outer boundary which is actually the sum of all the sides of a closed geometric object. A ruler is generally used to measure the sides of regular shapes whereas the string is used to measure the boundary of irregular shapes. As we know a triangle consists of 3 sides. Thus, the perimeter can be found by adding the length of all the three sides of a triangle.  

What is the Perimeter of a Triangle?

The perimeter of a triangle is the sum of the length of its 3 sides.


Perimeter of triangle (P) = Side1 + Side2  +Side3

Here a, b and c are the 3 sides of the triangle.

Thus, the perimeter of triangle (P) = a+b+c


Example: What is the perimeter of a triangle with sides of 3 inches, 4 inches and 5 inches?

Solution: Perimeter of triangle = Sum of all sides 

                                                  = (3 + 4 + 5) inches

                                                  = 12 inches

Therefore, the answer is 12 inches.

This principle is exactly the same for all triangles. 

Calculation of perimeter of a triangle:

On the basis of sides, triangles are divided into three types:

  1. Isosceles Triangle:  Two sides of the isosceles triangle are equal.

So, the perimeter of an Isosceles triangle = 2a + c


Example: In triangle ABC, 2 sides of the triangle (a) = 20 cm

Base (c) = 8 cm

Perimeter = 2(20) + 8

      = 48 cm


  1. Equilateral Triangle: All three sides of an equilateral are equal.

          So the perimeter = 3X


Where x is the length of each side of the triangle.

Example: Find the perimeter of an equilateral triangle of sides 2cm.

The perimeter is 3 ×2 cm = 6 cm


  1. Scalene Triangle: All sides of a scalene triangle are unequal.


       Perimeter = s +l + m

Example: In a given triangle ABC, find the perimeter for given triangle having three sides of 15 cm, 34 cm, 32 cm.


Therefore, perimeter = (15 + 34 + 32) cm = 81cm

On the basis of angles, triangles are divided into three types:

  1. Acute Triangle: A triangle with all the angles less than 90°. 

  2. Obtuse Triangle: A triangle with one angle of more than 90°.

  3. Right Triangle: A triangle with one angle equal to 90°.

   

Right Triangle - Out of all the three sides, one is the longest (hypotenuse). The other two is base and perpendicular depending on which side contains the taken angle.

To find out the perimeter of a right-angled triangle, add the length of both the legs (base and perpendicular) and then add the hypotenuse.

                     

So, the perimeter of right-angle triangle = a + b+ c

Where a and b are two legs of right angle and c is the opposite side of the right angle, also called as the hypotenuse.

If the lengths of the sides are not given, then Pythagoras Theorem is used to find out the length of the missing side. 

The theorem states that the addition of the square of both the sides of a right angle equals to the square of the hypotenuse.


The Pythagorean Theorem a2 + b2 = c2

If the value of any two sides (base and perpendicular) is known, then the perimeter can be written in terms of a and b only as follows: 

P = a + b + √ (a2 + b2)

Where a and b are the two legs of the triangle

Example 1:

Find the perimeter of a right-angle triangle with two legs of 6 cm and 3 cm. 

Let h be the hypotenuse, 

Then, h2 = 62 + 32.

        h2 = 36 + 9

        h2 = 45

Here, h must equal √45 which is equal to 6.71.

 Perimeter = 6.71cm + 6cm + 3cm = 15.71cm

So perimeter of the right triangle is 15.71cm.

Example 2:

Find the perimeter of a right-angle triangle with sides of lengths 4 ft and 3 ft. 

Solution 

Let in triangle ABC, side a = 4ft, b = 3ft, and h be the hypotenuse. 

Perimeter = a + b + h 

                = a + b + √(a2 + b2

                = 4 + 3 + √(42 + 32)

                = 12 ft


Example 3:

Find the perimeter of an isosceles right-angled triangle having a hypotenuse of 50 cm.

Solution

In isosceles right angled triangle, let a = b 

According to Pythagoras Theorem 

a2 + a2 = 502

2 a2 = 2500           (dividing both side by 2)

 a2 = 1250 

Perimeter = 2 a + 50 = 2 √1250 + 50 = 50(√ 2 + 1) =120.7 cm


Example 4:

Find the perimeter of an isosceles right-angled triangle, with sides of 8 cm.

Solution: By Pythagoras theorem,

Hypotenuse2 = (addition of two legs)2

Hypotenuse = √ (addition of two legs)2

= √2a2 = a√2 (a is the leg of triangle)

Perimeter of triangle = a + a + a√2 = 2a + a√2

= a (2 + √2) cm


Example 5:

The perimeter of an isosceles triangle is 26 cm. The length of each congruent side is 8 cm. find the length of the third side.

Solution

Perimeter = 8 + 8 + x

=>26 = 8 + 8 + x

=>26 = 16 + x

=>x = 26 -16 = 10 cm


Example 6:

An isosceles triangle with a height of 9cm has area of 90 cm2. What is perimeter of isosceles triangle?

Solution

From the area of the triangle, we can calculate the base.

Area = 1/2 (base).height

    90 = 1/2 ×b ×9

    90 = 4.5 ×b

      b = 20 cm

Now, to find the sides of an isosceles triangle, we will divide it into two right-angled triangles with base and perpendicular as 10 and 9 respectively. 


Hypotenuse of the right angled triangle(x2) = 92 + 102 

                                                                       = 81 + 100 

                              = 13.45cm

As the triangle is isosceles its two sides are equal.

So, perimeter of triangle = 20 + 13.45 + 13.45

                  = 46. 90 cm


Example 7:

The longest side of a triangle is twice as long as the shortest side. If the third side is 15 cm. Find two missing sides if the perimeter is 57 cm.

Solution:

Let a be one of the sides of the triangle

As per the question, the other side will be

b = 2a

c = 15 cm (given)

Perimeter = a + b+ c

            57 = a + 2a + 15 

            57 = 3a + 15 

      57- 15 = 3a 

             42 = 3a

      a = 42/3

                 a =14

b= 2a = 2 x 14 = 28

a= 14 cm, b = 28 cm