Calculate the Space Diagonal: Finding the Longest Distance Inside a 6-cm Cube

To find the length of the inner diagonal of a cube, we'll use the formula for the main space diagonal of a cube, which can be derived using the Pythagorean theorem:

The formula for the space diagonal ($d$) of a cube with edge length $a$ is:

$d = \sqrt{a^2 + a^2 + a^2}$.

Given that each side of the cube is 6 cm, substitute $a = 6$ cm into the formula:

$d = \sqrt{6^2 + 6^2 + 6^2} = \sqrt{3 \times 6^2} = \sqrt{3 \times 36} = \sqrt{108}$.

Now, calculate the square root of 108:

$\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}$.

Using a calculator or an estimated value for $\sqrt{3} \approx 1.732$, we calculate:

$6\sqrt{3} \approx 6 \times 1.732 = 10.392$.

Therefore, the length of the inner diagonal of the cube is approximately $10.39$ cm.

The correct choice for this problem is option 1: $10.39$ cm.