Question

Volume of HCP unit cell is :

• A. $24\sqrt{2}{r}^3$
• B. $8\sqrt{2}{r}^3$
• C. $16\sqrt{2}{r}^3$
• D. $24\sqrt{3}{r}^3$

Answer 1

The correct option is A $24\sqrt{2}{r}^3$

From the figure:
tan ${30}^0=\frac{a}{2\times y}$ or $y=\frac{a\times \sqrt{3}}{2\times 1}=\frac{\sqrt{3}}{2}a$
Area of hexagonal surface $=6[\frac{1a}{2}\times \frac{\sqrt{3}a}{2}]=\frac{6\sqrt{3}{a}^2}{4}$
Volume of hexagon $=$ Area of base $\times$ Height
$=\frac{6\sqrt{3}}{4}\times a^2\times 2\sqrt{\frac{2}{3}}a$
$=\frac{6\sqrt{3}}{4}\times {\left(2r\right)}^2\times 2\sqrt{\frac{2}{3}}\left(2r\right)=24\sqrt{2}{r}^3$