Will a 30cm antenna read Pet RFID chips over more than a few cm?
Building up a NFC or RFID antenna is not that simple. But, for a rule of thumb, the diameter of the antenna is the average reading distance.

To understand the challenges of building an antenna you need to know how it works. 

The antenna of the reader and the tag are a composition of coils which results in a complex RLC circuit. That circuits parameter depends on various of things like:


Shape of the antenna
number of windings
shape of windings
pcb
reader ic's parameters
etc...


The antenna then needs to be tuned to have it resonance on the carrier. For example: 13.56 Mhz. The quality of that RLC oscillating circuit must fit for the used modulation type. i.e. am or whatever is used.

I don't want to dive to deep into that, rather give you an idea of what is required for that rfid application.

Some reader-ic's are able to tune the antenna. If such a reader is used, it may help you with simple try-on-error approach.