The puzzle of the picky eater

Today’s Riddler post is a neat problem about calculating areas.

Every morning, before heading to work, you make a sandwich for lunch using perfectly square bread. But you hate the crust. You hate the crust so much that you’ll only eat the portion of the sandwich that is closer to its center than to its edges so that you don’t run the risk of accidentally biting down on that charred, stiff perimeter. How much of the sandwich will you eat?

Extra credit: What if the bread were another shape — triangular, hexagonal, octagonal, etc.? What’s the most efficient bread shape for a crust-hater like you?

Here is my solution:
[Show Solution]

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