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No: cyclohexane can exist in a few different conformations, none of which are planar. The most stable conformation is one that looks like a chair.
Other conformations include the boat and (more rarely - usually only as an intermediate) twisted boat conformations.The reason why cyclohexane in not flat is because its ring carbons are sp3 hybridised and hence they are at their lowest energy when the angle between substituents in 109.5°. In a perfectly flat hexagon the C-C angles would be 120°, thus incurring a hypothetical 11° of ring strain.
Therefore, by "puckering" into the chair conformation, cyclohexane attains the structure of lowest potential energy.
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