**Puzzle:**show the total area of the two semicircles is half the area of the large circle.

If you get stuck, go to +Alexander Bogomolny's wonderful website:

http://www.cut-the-knot.org/proofs/Semicircles.shtml

On top you'll get an applet that lets you slide the point where the semicircles touch - no matter where it is, the semicircles have the same total area! Click "hint" and you'll get a hint. If you're still stuck, scroll down and see a proof.

This puzzle is a lot harder than my other recent area puzzles. Indeed, it seems this fact was proved only in 2011!

• Andrew K. Jobbings, Two semicircles fill half a circle,

*The Mathematical Gazette*

**95**(Nov. 2011), 538-540.

I find that amazing, since people have been thinking about this stuff for millennia! However, Andrew Jobbings is a genius when it comes to 'recreational mathematics' - by which I mean math that's not considered 'serious', which people do just because it's fun. (This is a curious concept, now that I think about it.)

Check out more of his stuff here:

http://www.arbelos.co.uk/papers.html

#geometry