How many gallons of water does it take to fill a 15 foot round by 4 foot pool?

Here's a formula for round pools:

distance across*distance across*depth*5.9

for rectangles:

length*width*depth*7.5

or for a deep end:

length*width*((shallow end depth + deep end depth)/2)*7.5

so in your case, 15*15*4*5.9=5000 gallons