# HCP unit cell c-axis

Today I was asked to show that the perpendicular height, $$H,$$ of a regular tetrahedron is related to the length of a side, $$L,$$ of the tetrahedron, by $$H = \sqrt{(2/3)} \cdot L$$. I did this no problem.

I was then asked: Explain how this is relevant in determining a relationship between the unit cell parameters, $$a$$ and $$c$$ of a hexagonal-close packed crystal structure: $$c = 1.633 · a$$.

Could anyone possibly help me out here?

• Well, if you know what an HCP structure is, then you could have guessed that the said ratio of 1.633 was not put there on someone's whim. Feb 11 '20 at 19:17
• No I realise it’s twice of a - I think I know what’s going on but I don’t know what they’re asking for here Feb 11 '20 at 20:17
• Imagine the whole construction, with spheres and everything. That 1.633 is not some random decimal number. It must have been derived geometrically, you know, with square roots and stuff. Feb 11 '20 at 20:50