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12 inches = 1 foot. 240 inches = 20 feet. Use Pythagoras' theorem to find the diagonal of one of the faces of the cube. 2402+2402 = 115200. The square root of 115200 is 339.411255 Therefore the diagonal of the cube is 339.411255 inches or 339 inches correct to three significant figures Incidentally, by using the above answer and Pythagoras' theorem we can also work out the length of the longest internal diagonal within the cube which works out exactly as 340 inches.

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Q: What is the length in inches of a diagonal of a 20 foot cube?
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