How can I find the area of an overlayer structure?

What sources would you recommend (or if you could instead explain it to me that would be great). I have never studied crystallography but must do a module on it and in some of the questions we were given to practice the following is asked:

Give the area of a $$(\sqrt{3} \times \sqrt{3})$$R$$30^°$$ surface unit mesh on the surface of an (0001) hcp crystal with lattice parameters a = 4.2 Å and c = 5.5 Å?

• Is this homework?
– Camps
Nov 28, 2020 at 18:10
• This is a common task for overlayer structures, I have changed your title to make it more generally useful. This does look like homework, but this isn't something that is straightforward to look up. Nov 28, 2020 at 18:27
• It is not homework, but examples we were given and not contextualise in, and learning this during a pandemic where I haven't met the lecturer once, made this the only platform where I could try ask for help. Can someone indicate a book for me to learn this stuff? Nov 29, 2020 at 13:59

The 0001 facet area will be only dependent on the $$a$$ lattice constant. To solve for the area of that surface, you will just need to find the area of a rhombus with a $$60^{\circ}$$ angle.
To read the intended cell, start from the unit cell of the 0001 surface. Then multiply the surface vectors by the two values given, $$\sqrt{3}$$ and $$\sqrt{3}$$. Then you rotate the surface vectors around the z axis by the value given, $$30^{\circ}$$.
$$A = S^2\sin(A^{\circ})$$
$$A = (\sqrt{3}*4.2)^2\sin(60^{\circ})$$
$$A = 45.83$$