Question 44

The length of the diagonal of the cube is 6√3.

To find the diagonal of a cube with side length 6, we can use the formula for the diagonal \(d = a\sqrt{3}\), where \(a\) is the side length. Substituting in the value, \(d = 6\sqrt{3}\).

A) 6

This option represents only the length of one side of the cube, not the diagonal. The diagonal is a longer segment that spans across three dimensions, thus 6 is not the correct measurement.

B) 6√2

While this option is a common calculation for the diagonal of a square face of the cube, it does not apply to the full diagonal of the cube itself. The correct formula for the cube diagonal accounts for all three dimensions, making 6√2 insufficient.

C) 6√3

This is the correct answer. The length of the diagonal of a cube is derived from the formula \(d = a\sqrt{3}\). For a cube with a side length of 6, substituting gives \(d = 6\sqrt{3}\), accurately representing the diagonal's length.

D) 12

This option incorrectly suggests a linear measurement that does not correspond to the diagonal of the cube. The diagonal length is not simply the sum of the sides or any single dimension, making 12 an incorrect choice.

Conclusion

The length of the diagonal of a cube with side length 6 is definitively 6√3, as derived from the three-dimensional diagonal formula. Options A, B, and D fail to meet the criteria set by the geometric properties of a cube, confirming C as the only correct choice.