(i) Sounds like just another name for a cuboid:
Look at the special cases section.
(ii) Working on it.
LATER EDIT: (diagram in next post)
In my diagram the radius of the circumcircle is AF. If you let AC = CB = a then you can get an expression for the area in terms of a.
G is the centre of the incircle so its radius is BFtan(22.5) So you can get an expression for its area in terms of a.
As both will have a^2 this will cancel when you divide one by the other.
I've done that by measurement on my diagram and it doesn't appear to be any simple value.
Last edited by bob bundy (2014-02-01 19:45:19)