Worked example: Sodium is BCC with atomic radius r = 0.186 nm. Find its lattice constant a and theoretical density (atomic mass 23 u). Step 1: BCC relation: 4r = √3·a → a = 4×0.186/1.732 = 0.430 nm = 4.30×10⁻¹⁰ m. Step 2: Cell volume = a³ = (4.30×10⁻¹⁰)³ = 7.95×10⁻²⁹ m³. Step 3: Mass per cell = 2 atoms × 23 u = 46 × 1.661×10⁻²⁷ = 7.64×10⁻²⁶ kg. Step 4: Density = 7.64×10⁻²⁶ / 7.95×10⁻²⁹ ≈ 961 kg/m³. Step 5: Measured sodium density ≈ 971 kg/m³ — agreement within 1%! Conclusion: hard-sphere geometry predicts real densities — crystal structure is quantitative, not just pictures.