You have a square dartboard. What is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge?Go on. Try it now.
Here are some of my sketches as I attempted solutions.
Extension: Given an integer n>2, what is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge of a regular polygon with n sides? What is the probability as n approaches infinity?
I had a lot of fun working this out. Be sure to read Allison's original post on problem solving, persistence, and fearlessness.
If you just want to see the solution, you can find it at the bottom of the next post. In that post, I first find the solution to the Ngon Dartboard Probability (the extension offered above) followed by finding the solution to the instance when n=4 or square.
Square Dartboard Probability
Ngon Dartboard Probability
Ngon Dartboard Probability Simulation
Ngon Dartboard Probability Limit
UPDATE: Added conditions and changed wording.
UPDATE: Added links to the other posts.