To find the volume of a cube when given the length of its diagonal, let's go through the steps systematically.
1. Understand the Relationship:
- For a cube, the length of the diagonal (d) can be found using the formula:
[tex]\[
d = a\sqrt{3}
\][/tex]
where [tex]\( a \)[/tex] is the side length of the cube.
2. Given:
- The length of the diagonal of the cube is [tex]\( 4\sqrt{3} \)[/tex] cm.
3. Find the Side Length:
- We need to solve for [tex]\( a \)[/tex]:
[tex]\[
4\sqrt{3} = a\sqrt{3}
\][/tex]
- Divide both sides by [tex]\( \sqrt{3} \)[/tex]:
[tex]\[
a = \frac{4\sqrt{3}}{\sqrt{3}} = 4 \text{ cm}
\][/tex]
4. Calculate the Volume:
- The volume [tex]\( V \)[/tex] of a cube is given by:
[tex]\[
V = a^3
\][/tex]
- Substitute [tex]\( a = 4 \text{ cm} \)[/tex]:
[tex]\[
V = 4^3 = 64 \text{ cm}^3
\][/tex]
Therefore, the volume of the cube is [tex]\( 64 \text{ cm}^3 \)[/tex].
