The measurement that will require the use of irrational numbers is the diagonal of one of the cube’s faces.
To understand why, let's consider the diagonal of a face of the cube. The face of the cube is a square with side length \( a \). The diagonal \( d \) of the square can be calculated using the Pythagorean theorem:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]
Since \( \sqrt{2} \) is an irrational number, \( a\sqrt{2} \) will also be irrational unless \( a \) is zero.
In contrast, the surface area \( 6a^2 \), the volume \( a^3 \), and the area of one of the cube’s faces \( a^2 \) are all rational numbers if \( a \) is a rational number.
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