Let's analyze the measurements you mentioned for a cube with edge length \( a \):
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The diagonal of one of the cube's faces: The diagonal \( d \) of a face of the cube (which is a 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, the diagonal of one of the cube's faces will be an irrational number.
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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: \[ A = a^2 \] Since \( a \) is a rational number, \( a^2 \) is also rational.
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The surface area of the cube: The surface area \( SA \) of the cube is given by: \[ SA = 6a^2 \] Since \( a^2 \) is rational, \( 6a^2 \) is also rational.
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The volume of the cube: The volume \( V \) of the cube is given by: \[ V = a^3 \] Since \( a \) is rational, \( a^3 \) is also rational.
Conclusion:
The only measurement that requires the use of an irrational number is the diagonal of one of the cube's faces. All the other measurements (area of one face, surface area, and volume) yield rational numbers when \( a \) is a rational number.
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