If the area of an equilateral triangle is $24\sqrt{3}$ sq. cm, then its perimeter is

### Concept & Formula For an equilateral triangle of side $a$: Area $$ A = \frac{\sqrt{3}}{4}a^2 $$ Perimeter $$ P = 3a $$ ### Step-by-Step Solution 1. Given Area $= 24\sqrt{3}$. 2. $\frac{\sqrt{3}}{4}a^2 = 24\sqrt{3}$. 3. Divide both sides by $\sqrt{3}$: $\frac{a^2}{4} = 24$. 4. $a^2 = 96 \Rightarrow a = \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6} \text{ cm}$. 5. Perimeter $= 3a = 3(4\sqrt{6}) = 12\sqrt{6} \text{ cm}$. ### Exam Strategy & Shortcut Write $24\sqrt{3}$ as $\frac{\sqrt{3}}{4} \times (96)$. The side squared is 96, so the side is $\sqrt{96}$. Multiply by 3 for perimeter: $3\sqrt{96} = 3 \times 4\sqrt{6} = 12\sqrt{6}$. ### Common Pitfall Forgetting to take the square root of 96, or calculating the perimeter as just $4\sqrt{6}$ (which is the side, not the perimeter). ### Final Answer Therefore, the correct answer is **$12\sqrt{6}$ cm**.