To determine which measurement of the cube requires the use of irrational numbers, we can evaluate each option:
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The area of one of the cube’s faces: The area of a square face is calculated as \( a^2 \), which, since \( a \) is a rational number, results in a rational number.
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The diagonal of one of the cube’s faces: The diagonal \( d \) of a square face 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, the diagonal will be an irrational number when \( a \) is rational.
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The volume of the cube: The volume \( V \) of the cube is calculated as \( V = a^3 \), which results in a rational number if \( a \) is rational.
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The surface area of the cube: The surface area \( S \) is calculated as \( S = 6a^2 \), which results in a rational number if \( a \) is rational.
Based on this analysis, the measurement that will require the use of irrational numbers is:
The diagonal of one of the cube’s faces.