In a cube with an edge length \( a \), all of its edges are of equal length \( a \). The volume \( V \) of the cube is calculated using the formula:
\[ V = a^3 \]
Since \( a \) is a rational number, \( a^3 \) is also a rational number.
However, when you calculate the space diagonal \( d \) of the cube, you use the formula:
\[ d = a\sqrt{3} \]
In this case, since \( a \) is a rational number, \( \sqrt{3} \) is an irrational number. Therefore, the measurement that will require the use of irrational numbers is the space diagonal of the cube, \( d \).