Aha, initially I thought the question is asking about a distance between two points, vertices , which is not asked in this question. Cube's diagonal would be \(d=side*{\sqrt{3}}\).

Here, surface distance is the distance on the surface of cube's sides. Like, we are drawing a line on the sides of a cube shaped box. Therefore, the required distance is a diagonal between two sides and the side itself \(side*{\sqrt{2}}+side\). Answer is \(6*{\sqrt{2}}+6\), and it's indeed A.

Thanks for posting a good question.

Explanation:

The key thing to notice in this question is - "the shortest path on the surface of the cube"

Refer to the figure below,
Surface distance = \(\sqrt{12^2 + 6^2} = \sqrt{144 + 36} = \sqrt{180} = 6\sqrt{5}\)

Hence, option A

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very useful question for analysis of geometry