Area of Isosceles Triangle

Example 1: A shelf has a shape of an isosceles triangle that has a base of 16 cm and height of 11 cm. What is the area of the shelf?

Solution:
Base of the triangle (b) = 16 cm
Height of the triangle (h) = 11 cm

Area of Isosceles Triangle = \(\frac{1}{2}\) x b x h

⇒ \(\frac{1}{2}\) x 16 x 11

⇒ 8 x 11

⇒ 88 cm\(^2\)

So, the area of the shelf is 88 square centimeters.

Example 2: The base and height of an isosceles triangle ‘A’ are one half of the base and height of an isosceles triangle ‘B’. How many times greater is the area of triangle ‘B’ as compared to triangle ‘A’.

Solution:
Consider triangle ‘B’ :
Let the base be the ‘\(b\)’ units and height as ‘\(h\)’ units

Area of triangle ‘B’ = \(\frac{1}{2} \times b \times h = \frac{bh}{2}\) square units.

Consider triangle ‘A’ :

According to the given statement, the base is ‘\(\frac{b}{2}\)’ units and height is the ‘\(\frac{h}{2}\)’ units.

Area of triangle ‘A’ = \(\frac{1}{2}\times\frac{b}{2}\times\frac{h}{2}=\frac{bh}{8}\) square units

\(\frac{\text{Area of triangle ‘B’}}{\text{Area of triangle ‘A’}} = \frac{\frac{bh}{2}}{\frac{bh}{8}}\)

\(\frac{\text{Area of triangle ‘B’}}{\text{Area of triangle ‘A’}} = \frac{4}{1}\)

Area of triangle ‘B’ is four times the area of triangle ‘A’.

Example 3: Find the height and area of the isosceles triangle.

Solution:

Height (h) of isosceles triangle = \(\sqrt{a^2-\frac{b^2}{4}}\)

⇒ \(\sqrt{144-16}\)

⇒ \(\sqrt{128}\)

⇒ 11.31 cm

Height of the isosceles triangle is 11.31 cm

Area of the isosceles triangle,

Area (A) = \(\frac{b}{4}\sqrt{4a^2-b^2}\)

⇒ \(\frac{8}{4}\sqrt{4{\times12}^2-8^2}\)

⇒ \(2\sqrt{576-64}\)

⇒ \(2\sqrt{512}\)

⇒ \(2\times22.62\)

⇒ 45.25 cm\(^2\)

So, the area of an isosceles triangle is 45.25 square centimeters.