What's the Probability a Circle is Entirely in the Square? 02/24 Jane Street Monthly Puzzle

A circle is randomly generated by sampling two points uniformly and independently from the interior of a square and using these points to determine its diameter. What is the probability that the circle has a part of it that is off the square? Give your answer in exact terms.
This is the Jane Street Monthly Puzzle in February 2024.
Problem Link: www.janestreet...
Solution and Leaderboard Link: www.janestreet...

[@lyricsworld320]

Amazing explanation

[@obergaba-sj4kb]

can you do the jane street puzzle of march?

[@luckyleeds]

I solved the march puzzle by working around the top right 31 cell and trying different combinations for the outermost 2 L-hooks, since there are only a few combinations to make up 31. If I figure out a more efficient trick to solve it I will make a video.