Which Beach is Closer?

Published on March 18, 2025

The Problem

You are planning a picnic on the remote tropical island of 𝜋-land. The island’s shape is a perfect semi-disk with two beaches, as illustrated below: Semicircular Beach (along the northern semicircular edge of the disk) and Diametric Beach (along the southern diameter of the disk).

Problem

If you pick a random spot on 𝜋-land for your picnic, what is the probability that it will be closer to Diametric Beach than to Semicircular Beach? (Unlike the illustrative diagram above, assume the beaches have zero width.)

My Approach

To find the solution I did the part that sounded the most interesting, finding the shape of the line of delineation

Solution

I did not find the exact ratio, nor did I find the exact line. These would be fun to do (but later)

I have made the code used for this available here