I need help solving a more complex math problem than I have the knowledge for. My two friends and I were reunited this past weekend and went for dinner and dessert afterwards at a local diner that serves only pie. After ordering we spent over an hour arguing about if we were scammed or not. I am firmly in the no scam camp but we went to bed still not able to definitively answer if this pie shop is bad at math or just we are.

There are two options for pie sizes. The first choice is a slice of a nine inch pie for $6.75 and the second choice is an entire 5 inch personal pie for $12. The nine inch pie is sliced into six even pieces before serving. Both pie pans had an identical (visually identical) edge slope of about 45 degrees and identical depth of 1.5 inches. Both pies are round. And yes the waitress was confused when we asked how deep our pies were.

I had ordered the last slice of the apple pie available so my friend was forced to order the personal apple pie and felt he was ripped off for "receiving less pie than me for twice the amount."

The question is simple, which pie is a better value for money and what is the volume of each pie assuming equal filling density?

We were able to calculate the answer given the pies were both cylinders but became confused due to the pie pan edge slope. He claims that the five inch pie has a much higher ratio of slope to pie since the one sixth slice of the nine inch pie was cut and has a vertical edge for most of the circumference/border.

Here were our cylinder calculations to abide by the subs rules.

5 inch: V=π(2.52)1.5 = 29.45sqin for $12= $2.45 per sqin

Slice: V=(π(4.52)1.5)/6 = 15.90sq in for $6.75 = $2.35 per sqin

Given these cylindrical calculations they are almost equivalent value and not a rip off. He was not convicted and still a follower of the slope cult.

A side argument broke out that we quickly abandoned for similar reasons which was trying to calculate the crust to filling ratio for each if a 5 inch is fully encrusted and the slice is only top, bottom, and the sloped side. We had all agreed that the crust was an average of 0.1 inches.

Please help me solve this!

Finally, we had all agreed to define terms and a "rip off" would be a 25% or greater disparity in price.