A cube has side length 6. What is the length of the diagonal of the cube?

Rationale

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

To find the length of the diagonal in a cube, we can use the formula (d = s sqrt{3}), where (s) is the length of a side. For a cube with a side length of 6, the diagonal calculates to (6 sqrt{3}).

A (6) YOUR ANSWER

This choice represents the length of one side of the cube, not the diagonal. The diagonal spans from one vertex of the cube to the opposite vertex, which is longer than the individual side lengths.

B (6√2) YOUR ANSWER

This value correctly represents the length of the diagonal across one face of the cube, not the space diagonal that connects opposite corners. The diagonal across a face would be calculated as (s sqrt{2}), which does not apply to the entire cube.

C (6√3) CORRECT

This is the correct calculation for the diagonal of the cube. Using the formula (d = s sqrt{3}), substituting (s = 6) results in a diagonal length of (6sqrt{3}). This length is derived from the three-dimensional nature of the cube.

D (12) YOUR ANSWER

This choice does not correspond to any relevant measurement in this context. The diagonal of the cube cannot be simply the sum of the side lengths or any other direct relation to the sides. It results from applying the correct geometric formula.

Conclusion

The diagonal of a cube is determined by the formula (d = ssqrt{3}), where (s) is the side length. For a cube with a side length of 6, the diagonal measures (6sqrt{3}), distinguishing it from the lengths of sides or face diagonals. Understanding this relationship is essential in geometry, especially when dealing with three-dimensional shapes.

A cube has side length 6. What is the length of the diagonal of the cube?

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

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