Area of Isosceles Triangle

Area of Isosceles Triangle

There are three types of a triangle based on the length of the sides. They are as follows:

1.  Equilateral triangle

2.  Isosceles triangle, and 

3. Scalene triangle.

Equilateral Triangle: An equilateral triangle is a triangle whose all three sides are equal and each interior angle measures 60o.

Isosceles Triangle:  An isosceles triangle is a triangle whose two sides are equal.

Scalene Triangle:  A scalene triangle is a triangle whose all three sides are unequal.

Area of Isosceles Triangle

The area of an isosceles triangle is the amount of space that it occupies in a 2-dimensional surface.

So, the area of an isosceles triangle can be calculated if the length of its side is known.

Let us consider an isosceles triangle whose two equal sides length is ‘a’ unit and length of its base is ’b’ unit. Then, 

Derivation of area of an Isosceles Triangle

The formula for the area of an isosceles triangle can be derived using any of the following two methods.

1. Using basic area of triangle formula

2. Using Heron’s formula

METHOD: 1 

Deriving area of an isosceles triangle using basic area of triangle formula

Since, the altitude of an isosceles triangle drawn from its vertical angle bisects its base at point D.

So,

We can determine the length of altitude AD by using Pythagoras theorem.

In ∆ADC, Right angled at angle D. Then, hypotenuse = AC, altitude = AD and base = DC.

According to Pythagoras theorem,

METHOD: 2 

Deriving area of an Isosceles Triangle Using Heron’s Formula

Solved Examples:

Q.1. Find the Perimeter and Area of an Isosceles Triangle Whose two Equal Sides and Base Length is 5 cm and 6 cm Respectively.

Ans.   Given, length of two equal sides of an isosceles triangle = a = 5 cm 

          And length of its base = b = 6 cm

Perimeter of an isosceles triangle = 2a + b 

                                                          = 2(5) + 6

                                                           = 10 + 6 = 16 cm

Q.2. If the Base and Area of an Isosceles Triangle are 8 cm and 12 cm2 respectively. Then find its perimeter.

Ans.  Given, length of base = b = 8 cm

                      And, area = 12 cm2 

Q.3. Find the Altitude of an Isosceles Triangle Whose Two Equal Sides and Base Length is 7 cm and 4 cm Respectively.

Ans. Given, length of two equal sides of an isosceles triangle = a = 7 cm 

          And length of its base = b = 4 cm

Q.4. Find the Area of Right Isosceles Triangle Whose Hypotenuse is 5\[\sqrt{2}\] cm.

Ans.  Let the two equal sides AB and BC of the right isosceles triangle ABC be ‘a’ cm each and AC be the hypotenuse of length 5\[\sqrt{2}\] cm.

                                                                                                                    = 12.5 cm2