Prove that the volume of hexagonal of the unite cell =0.866(a seqer).c ?

I'm not sure what your question is.
What is "hexagonal of the unite cell"
What does "(a seqer)" mean?
What field of study does this relate to?

This sounds like crystallography, so it's a hexagonal prism, where c is the atomic distance.

OK. I suspected something like that.
I looked at Crystal structure - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Crystal_structure)
it looks like you need c and a from the picture there unless we're assuming that a=c. The picture says a!=c but I assume they mean that a does not have to equal c. I'll go with that until the OP tells me otherwise.

Volume of a hexagonal prism is area of base * height. Height = c. base is hexagon and area of regular hexagon is \frac{3 \sqrt{3}}{2}a^2 . Therefore since we're assuming that a=c the volume is \frac{3 \sqrt{3}}{2}c^3 but that = 2.598c^3 which is not what the OP wrote. I guess we'll have to wait for him to supply more info and also tell us if he need to include proof of the area of a hexagon.

It's actually a third of your answer, asterisk. If you look at the hexagonal diagram in the wikipedia article. You will see that the hexagonal cell is made up of 3 identical rhombic prisms (or at least it would be if it were tessellated). THESE are the unit cell of a hexagonal lattice. So there's your 0.866. I don't know what seqer means though.

I'll let this one slide since you're not into crystallography ;)

OK. Sorry :o . My 1.5 minutes of study must have missed that fine point :p

So the question is really "what is the volume of the unit cell in a hexagonal crystal structure?"? And the answer would require knowledge that the unit cell of hexagonal crystal structure is really a rhombic prism?

I need the correct answer 4 this question ***

Asma - see: https://www.askmehelpdesk.com/math-sciences/prove-volume-hexagonal-0-866a-2c-567394.html#post2762743