Which of the following is the correct relationship between edge length (a) and radius of sphere (r) located in a fcc unit cell?

All of these

The correct answer is option 4. All of these.

Face centered cubic unit cell (FCC)

let the unit cell edge length be ‘a’ and face diagonal AC = b.

In ∆ ABC

AC² = b² = BC² + AB² = a² + a² = 2a² or b = \(\sqrt{2}\)a

If r is the radius of the sphere, we find b = 4r = \(\sqrt{2}\)a or a = \(\frac{4r}{\sqrt{2}}\) 

a = 2\(\sqrt{2}\)r

we can also write, r = \(\frac{a}{2\sqrt{2}}\)