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The largest square that you can fit in any cube will have each corner 1/4 of the way away from a corner of the cube and will have a side length of s*3√2/4 where s is the length of the sides of the cube. For a unit cube, the side length is 1 by definition, meaning that this largest square will have a side length of 3√2/4 and that its area will be 9/8 units squared.

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Q: In a unit cube what is the largest square you can have inside?
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