a cube has an edge linked a that is rational number of centimeters identify which measurement will require the use of irrational numbers. Would it be the surface area of the cube be diagonal one of the cubes faces the volume of the cube or the area of one of the cubes faces.

1 answer

To analyze which measurement will require the use of irrational numbers, let's examine each of the options given the edge length \( a \), where \( a \) is a rational number:

  1. Surface Area of the Cube: The surface area \( S \) of a cube is given by the formula: \[ S = 6a^2 \] Since \( a \) is a rational number, \( a^2 \) is also rational, and thus \( S \) will be rational.

  2. Diagonal of One of the Cube's Faces: The diagonal \( d \) of a face of the cube 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 \( d \) will be irrational when multiplied by the rational number \( a \).

  3. Volume of the Cube: The volume \( V \) of a cube is given by the formula: \[ V = a^3 \] Since \( a \) is rational, \( a^3 \) will also be rational.

  4. Area of One of the Cube's Faces: The area \( A \) of one face is given by: \[ A = a^2 \] As \( a \) is rational, \( A \) will also be rational.

Conclusion:

Out of the options given, the measurement that will require the use of irrational numbers is the diagonal of one of the cube's faces.