Let's analyze the measurements you mentioned for a cube with edge length
-
The diagonal of one of the cube's faces: The diagonal
of a face of the cube (which is a square) can be calculated using the Pythagorean theorem: Since is an irrational number, the diagonal of one of the cube's faces will be an irrational number. -
The area of one of the cube's faces: The area
of a face of the cube is given by: Since is a rational number, is also rational. -
The surface area of the cube: The surface area
of the cube is given by: Since is rational, is also rational. -
The volume of the cube: The volume
of the cube is given by: Since is rational, 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