I'd like a homework pizza with everything on it, but no loops
HTML/JavaScript, 160 bytes

(linebreaks/indentation only added here do prevent scrolling)

&lt;input onchange='window.n++||(n=1,L=1e9);s=this.value.split(" ");d=s[2];
        a=[5.5/7*d*(s[3]||d),d*d][1*(s[1]=="s")];c=s[0]/a;alert(a!=a?m:a+"@"+c);
        L&lt;c||(L=c,m=n)'&gt;


The input is a text box (press Enter to submit); the output is an alert box.

A few caveats (or cheats, if you will):


Pi/4 is 5.5/7
in some cases, it assumes that the pizzas have non-zero areas
it's not possible to enter the same pizza twice in a row (because that won't trigger a change event)
note that defocusing the input also counts as "enter"
assumes that at least one pizza costs less than a billion Drachmas per square attoparsecs
no fault tolerance for the input whatsoever
you could argue that it's not qualified because it uses the browser's event loop