I'm not sure I understand sloped armour.
Okay, sloping the armour increases its effective thickness against a horizontal shot, which is Good. But sloping a plate also reduces its vertical height - if you want the sloped plate to extend to same height as a non-sloped plate then it needs to be longer, and the weight gain from the longer plate exactly cancels out effective-thickness increase gained from sloping.
But sloping the armour allows variations in the tank shape and its internal volume, taking it from a cube towards a pyramid with a less obvious profile... however, you also reduce the internal volume which makes it harder to fit inside important things like the crew and the engine, unless you make the tank broader or longer... maybe this is why it wasn't simple to design tanks with sloped armour, particularly not as a derivative of a current design?
But there's a further issue here that confuses me - battleships. The KGVs used thick external vertical belts, while the South Dakotas used inclined belts. Assuming that you want the same vertical height of armour whether you use an inclined or vertical belt, then there is no weight saving - the inclined belt has a greater effective thickness and hence can be thinner, but the weight saved is immediately lost because the plate needs to be higher. In this case how can there be an internal volume argument because the inclined belt was already internal...
Hmm another thing. Looking at the South Dakota wiki page says that it has a belt thickness of 12.2", inclined at 19 degrees, giving an effective thickness of 17.3". But a 12.2" plate inclined at 19 degrees has a horizontal thickness of only 12.2/cos 19 = 12.9".
This makes me think that the "effective-thickness" calculation that I've been using is fundamentally incorrect and that the benefits of sloping are described by a different formula. Can anyone explain please?