Calculating c/a ratio for h.c.p

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SUMMARY

The ideal c/a ratio for a hexagonal close packing (HCP) unit cell is definitively established as 1.633. The discussion emphasizes that the atomic packing factor is not relevant for this calculation; instead, the solution lies in understanding the geometry of the HCP structure. Participants suggest visualizing the unit cell with spherical atoms and considering the centroid of an equilateral triangle to derive the ratio accurately.

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mate1000
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iv been asked to show that the ideal c/a ratio for a hexagonal close packing unit cell is 1.633. The only thing i could come up with was to do with the atomic packing factor. where i get (4/3pi^2r*6) / (3a*2r*c). Am i heading in the right direction or am i way off course. If I am way off would anyone be able to give me a hand?
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The packing factor isn't useful. This is simply a question of 3d geometry. Draw a picture of the HCP unit cell with spherical atoms and start there.

Hint: What do you know about the position of the centroid of an equilateral triangle?
 

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