To determine which measurement will require the use of irrational numbers, we analyze the options provided in relation to a cube with edge length \( a \).
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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, this measurement requires the use of irrational numbers.
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The area of one of the cube’s faces: The area \( A \) of a face of the cube is given by: \[ A = a \times a = a^2. \] This is a rational number since \( a \) is rational.
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The volume of the cube: The volume \( V \) of the cube can be calculated as: \[ V = a^3. \] This is also a rational number because \( a \) is rational.
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The surface area of the cube: The surface area \( SA \) of the cube is calculated as: \[ SA = 6a^2. \] This is a rational number since it is based on \( a^2 \) which is rational.
Based on this analysis, the only measurement that requires the use of irrational numbers is:
the diagonal of one of the cube’s faces.