diameter of a 16 space hypercube

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By "diameter" I assume you mean the longest measurement, corresponding to the long diagonal (the one that goes through the center, not along one side;top to bottom and front to back and left to right) of a 3D cube.

We look at the known cases and generalize:
Using regular figures of side s

Square, 2D:
Length of diagonal = sqare root of (s^2 + s^2) = root2 * s

Cube, 3D:
Length of long diagonal = square root of (s^2 + s^2 + s^2) = root3 * s

Generalizing to n ...