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If cyclohexane were planar, every three consecutive carbons would share a 120º angle, very inadequate for a sp3 hybridization. Besides, all hydrogens would be eclipsed.

Due to that, cyclohexane leaves planarity.

Cyclohexane is not planar but adopts a so-called chair conformation, where every three consecutive carbons bear an angle very close to 109.5º, the natural angle of the sp3 hybridization.
In addition, all hydrogens are perfectly alternated in the chair conformation.

Can you tell apart two different hydrogen positions in cyclohexane's chair conformer?
Yes, equatorial (E) and axial (A).

Since cyclohexane is built from single C-C bonds, their rotation is feasible. The combined rotation of various bonds in cyclohexane gives rise to a conformational equilibrium where the chair inverts.
Please note that the chair inversion produces an exchange of positions: all ecuatorial hydrogens move to be axial and viceversa.

The mechanism of combined bond torsion is quite complex and implies the passing through 'boat' and 'twist boat' conformers, with a global inversion barrier of 11 kcal/mol, four time as much as in ethane.
However bigger is the barrier in cyclohexane, the chair inversion happens several thousands times per second.

it is like drawing two 'siamese' butane molecules sharing the two methyls turned into CH2 groups.
Look how an axial-methyl bearing cyclohexane is drawn.