Perimeter of an Isosceles Triangle

A triangle can be considered as an isosceles triangle if and only if two sides of the triangle have the same length and two equal angles. The perimeter of an isosceles triangle comes under the parent topic mensuration which is a branch of geometry that deals with measurements of 2D/3D figures.

Perimeter of Isosceles Triangle

The perimeter of an Isosceles Triangle can be calculated by adding lengths of all three sides. In the isosceles triangle, two sides are having an equal length. So we can calculate the perimeter if the base length and other two equal side lengths are known. 

The formula of the perimeter of an isosceles triangle is given by-

Perimeter of Isosceles Triangle = 2a + b

Where 

a is length of two equal sides.

b is length of base (Unequal side).

Perimeter of Isosceles Right Angled Triangle

The perimeter of an isosceles triangle is the sum of all lengths of sides. In an isosceles right-angled triangle, one unequal length side is hypotenuse which is the larger side. The perimeter is given by formula-

Perimeter of Isosceles right angled Triangle = h + 2l

Where

h is length of hypotenuse.

l is length of equal side.

Apply Pythagoras theorem 

h² = l² + l²

h = √(l² + l²)

h = √(2l²)

h = √2l (or) l = h/√2

So even knowing either of hypotenuse or length of another side we can still find out the perimeter.

So when we have hypotenuse length then,

Perimeter = 2(h/√2) + h

where h is length of hypotenuse.

 If we have the length of equal sides then

Perimeter = 2l + √2l

where l is length of equal sides.

Let’s look into a few examples to get more understanding

Sample Problems

Question 1: What is the perimeter of an isosceles triangle if its side lengths are 3cm, 5cm, 5cm.

Solution:

For an isosceles triangle two side length are same

So length of two equal  sides (a) = 5cm

Base length (b) = 3cm

Perimeter of isosceles triangle = 2a + b

                                               = 2(5) + 3

                                               = 10 + 3

                                               = 13cm

So perimeter = 13cm

Question 2: What is the perimeter of an isosceles triangle if its base length is 5cm and the equal side length is 8cm.

Solution:

For an isosceles triangle two side length are same

So length of two equal  sides (a) = 8cm

Base length (b) = 5cm

Perimeter of isosceles triangle = 2a + b

                                                = 2(8) + 5

                                                = 16 + 5

                                                = 21cm

So perimeter of given isosceles triangle is 21cm.

Question 3: What is the perimeter of a right-angled isosceles triangle if its hypotenuse length is 7cm and the equal side length is 4cm.

Solution:

Given

Equal sides length (l) = 4cm

Hypotenuse length (h) = 7cm

Perimeter of right angled isosceles triangle = 2l + h

                                                                    = 2(4) + 7

                                                                    = 8 + 7

                                                                    = 15cm

So perimeter for a given triangle = 15cm

Question 4: What is the perimeter of a right-angled isosceles triangle if its hypotenuse length is 6cm.

Solution:

Given

Hypotenuse length (h) = 7cm

We know the perimeter of isosceles right angled triangle when hypotenuse is given is

Perimeter = 2(h/√2) + h

                = 2(7/√2) + 7

                = 7√2 + 7

                = 16.9

So perimeter for a given triangle is 16.9cm

Question 5: What is the perimeter of a right-angled isosceles triangle if it’s one side length is 5cm.

Solution:

Given

Side length (l) = 5cm

We know the perimeter of isosceles right angled triangle when length of a side is given 

Perimeter = 2l + √2l

                = 2(5) + √2(5)

               = 10+5√2

               = 17.07

So perimeter for a given triangle is 17.07cm