Correct Answer: 154

Solution :
Given that the length of each side of an equilateral triangle is $14 \sqrt{3}\ cm$
BD = DC = $7\sqrt3$ cm
By Pythagoras theorem,
$AD=\sqrt{AB^2-BD^2}$
$= \sqrt{(14\sqrt3)^2-(7\sqrt3)^2}$
= 21 cm
Since the radius of the incircle is one-third of the height of the equilateral triangle,
OD = radius of incircle = $\frac{1}{3}\times$ 21 = 7 cm
So, the area of incircle = $\pi r^2=\frac{22}{7}\times 7^2=154$ cm ²
Hence, the correct answer is 154.