Question:
Suppose $PABCD$ is a right square pyramid with apex $P$ and base $ABCD$.  If $PBD$ is an equilateral triangle with side length 6, then what is the volume of $PABCD$?

Answer:
[asy]

import three;

triple A = (0,0,0);

triple B = (1,0,0);

triple C = (1,1,0);

triple D = (0,1,0);

triple P = (0.5,0.5,1);

draw(B--C--D--P--B);

draw(P--C);

draw(B--A--D,dashed);

draw(P--A,dashed);

label("$A$",A,NW);

label("$B$",B,W);

label("$C$",C,S);

label("$D$",D,E);

label("$P$",P,N);

triple F= (0.5,0.5,0);

label("$F$",F,S);

triple M=(B+C)/2;

draw(D--B,dashed);

draw(P--F,dashed);

[/asy]

Let $F$ be the center of the square base.  Since the pyramid is a right pyramid, segment $\overline{PF}$ is an altitude of triangle $PBD$.  Since $PBD$ is an equilateral triangle with side length 6, $PFB$ is a 30-60-90 triangle with $FB = BD/2 =3$ and $PF = 3\sqrt{3}$.  Finally, $\overline{BD}$ is a diagonal of square base $ABCD$, so we have $BC = BD/\sqrt{2} = 6/\sqrt{2} = 3\sqrt{2}$.  Therefore, the volume of the pyramid is \[\frac{[ABCD](PF)}{3} = \frac{(3\sqrt{2})^2 (3\sqrt{3})}{3} = \boxed{18\sqrt{3}}.\]