The volume (in cm$^3$) of a cube, whose principal diagonal measures $4\sqrt{3}$ cm, is :

Question

Accepted Answer

To determine the volume of a cube whose principal diagonal measures $4\sqrt{3}$ cm, we need to use the relationship between the side length of the cube and its diagonal.

1. **Understanding the Relationship:**
   - For a cube with side length $a$, the formula for the principal diagonal $d$ is given by:
     $$
     d = a\sqrt{3}
     $$
   - We are given that the principal diagonal $d = 4\sqrt{3}$ cm.

2. **Finding the Side Length:**
   - Using the formula for the diagonal, we can set up the equation:
     $$
     a\sqrt{3} = 4\sqrt{3}
     $$
   - To solve for $a$, divide both sides by $\sqrt{3}$:
     $$
     a = 4
     $$

3. **Calculating the Volume:**
   - The volume $V$ of a cube is given by the formula:
     $$
     V = a^3
     $$
   - Substituting the side length $a = 4$ into the formula:
     $$
     V = 4^3 = 64 \text{ cm}^3
     $$

4. **Justification for Option 3:**
   - The calculated volume of the cube is $64$ cm$^3$, which matches Option 3. Therefore, Option 3 is the correct answer.

5. **Explanation for Incorrect Options:**
   - **Option 1 (8 cm$^3$):** This would imply a side length of $a = 2$, which would result in a diagonal of $2\sqrt{3}$, not $4\sqrt{3}$.
   - **Option 2 (27 cm$^3$):** This would imply a side length of $a = 3$, which would result in a diagonal of $3\sqrt{3}$, not $4\sqrt{3}$.
   - **Option 4 (125 cm$^3$):** This would imply a side length of $a = 5$, which would result in a diagonal of $5\sqrt{3}$, not $4\sqrt{3}$.

In conclusion, the correct volume of the cube, given the principal diagonal of $4\sqrt{3}$ cm, is indeed $64$ cm$^3$, which is why Option 3 is the correct answer.