Using a sphere as an example.

* surface area = 4 * PI() * r * r

* armor volume is given

* sphere volume = 4/3 * PI() * r * r * r

So:

outer radius = cube root of ( 3 * hull_volume / ( 4 * PI() ) )

and

volume of space not used by armor (vi) = hull volume - armor volume

and then:

radius of space not used by armor (ri) = cube root of ( 3 * vi / ( 4 * PI() )

and finally:

thickness of armor = outer radius - inner radius

**Represented as one big ugly messy equation:**

thickness of armor =

cube root of ( 3 * hull_volume / ( 4 * PI() ) )

- cube root of ( 3 * (hull volume - armor volume) / ( 4 * PI() )