A cube has an edge length, a , that is a rational number of centimeters. Identify which measurement will require the use of irrational numbers. (1 point) Responses the area of one of the cube’s faces the area of one of the cube’s faces the diagonal of one of the cube’s faces the diagonal of one of the cube’s faces the surface area of the cube the surface area of the cube the volume of the cube

1 answer

To determine which measurement will require the use of irrational numbers for a cube with a rational edge length \( a \):

  1. Area of one of the cube’s faces: The area of a face is calculated as \( a^2 \), which remains a rational number.

  2. Diagonal of one of the cube’s faces: The diagonal \( d \) of a face can be found using the formula \( d = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, the diagonal of the face will be irrational if \( a \) is rational.

  3. Surface area of the cube: The surface area is given by \( 6a^2 \), which is rational since it is based on the square of \( a \).

  4. Volume of the cube: The volume is calculated as \( a^3 \), which is also rational.

Thus, the measurement that will require the use of irrational numbers is the diagonal of one of the cube’s faces.