The unit cell of diamond is a cube with an edge of 356.7 pm, and it contains 8 carbon atoms…

Step 1. Mass of the cell. $8 \times 12.011 / (6.022 \times 10^{23}) = 1.5956 \times 10^{-22}\ \text{g}$.

Step 2. Volume of the cell. $(3.567 \times 10^{-8}\ \text{cm})^3 = 4.538 \times 10^{-23}\ \text{cm}^3$.

Step 3. Density. $1.5956 \times 10^{-22} / 4.538 \times 10^{-23} = 3.516\ \text{g/cm}^3$.

Step 4. Bond length. $356.7 \times \sqrt{3}/4 = 154.5\ \text{pm}$.

Step 5. Space filled. The radius is $77.2\ \text{pm}$. Eight atoms fill $8 \times \dfrac{4}{3}\pi\,(77.2)^3 = 1.543 \times 10^{7}\ \text{pm}^3$. The cell is $(356.7)^3 = 4.539 \times 10^{7}\ \text{pm}^3$. The fraction is $0.340$, which is $34.0\%$.