In the context of a cube with edge length \( a \) (where \( a \) is a rational number), there are several measurements we can consider, but certain derived measurements may require the use of irrational numbers.
-
Surface Area: The surface area \( SA \) of a cube is given by the formula: \[ SA = 6a^2 \] Since \( a^2 \) is also rational (the square of a rational number is rational), the surface area will be rational.
-
Volume: The volume \( V \) of a cube is given by the formula: \[ V = a^3 \] Similarly, \( a^3 \) is also rational, so the volume will be rational.
-
Diagonal of the Cube: The space diagonal \( d \) of a cube can be calculated using the formula: \[ d = a\sqrt{3} \] Since \( \sqrt{3} \) is an irrational number, the diagonal \( d \) will be irrational when \( a \) is a rational number.
Thus, the measurement that requires the use of irrational numbers when the edge length \( a \) is a rational number is the space diagonal of the cube.