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1) Hexagonal closest packing (efficiency = 74%)

2) Cubic closest packing - (efficiency = 74%)

3) Body-centered - (efficiency = 68%)

I know the efficiency values because they are in my textbook and all over the internet. I need to come up with these numbers myself.

efficiency = Volume of the spheres/Volume of a box around the spheres.

the spheres in my model have a radius of 3cm, so volume of the spheres is easy... 4/3(pie)r^3 x (number of spheres)

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**I'm trying to figure out a way to find the volume of a box around Hexagonal closest packing and cubic closest packing.**

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**Hexagonal closest packing**: 13 balls in total in 3 layers. (see pictures below)

bottom layer: 3 spheres

middle layer: 7 spheres

top layer 3 spheres

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**Cubic closest packing**: 13 balls in total in 3 layers

bottom layer: 4 balls

middle layer: 5 balls

top layer: 4 balls

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I know that the volume of both must be the same since they each have 13 balls and each have the same packing efficiency.

How do I find the volume of a box around these spheres?

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I've been trying to figure this out for hours so any help at all is appreciated........and sorry for the length of the question