TxO Math Bowl 2024 - Guts Contest - Set 8 Problem 3

Munch wants to scam people in a carnival game! However, all the people are smart, so he has to make the game fair (by fair, the expected amount of money Munch makes is 0). Jerry, being bored, decides to play the game. The game is played as shown below:

1. Jerry chooses a polygon (or line) \Gamma.
2. Jerry pays some amount of money to Munch.
3. Jerry throws the polygon onto an infinite square grid of horizontal lines spaced 1 unit apart and vertical lines spaced 1 unit apart.
4. Munch pays Jerry the number of intersections of \Gamma with the lines on the grid.

First, Jerry gives Munch an equilateral triangle with side length 1, and pays \frac{6}{\pi}. Then, Jerry gives Munch a regular octagon, and pays \frac{32}{\pi}. If the area of the octagon is K, find \lfloor K \rfloor.