Circle in the Square, in the Circle
There is a hard problems at the bottom of this post.
I like my students to be prepared to confront strange, hard, math problems. In combinatorics, each test is worth 90 points, with an extra 20 points at stake for attacking a previously unfamiliar problem. (Yes, I allow more than 100 points, but note, 20 points are on things they weren’t taught.) On the “new” problem I am not looking for a correct answer (though I wouldn’t mind them), but good process. The kids have to demonstrate Polya-style problem solving, with explicit reference to what they are trying, and why, and what happens. They should know to try diagrams, look for similar problems, break problems into chunks, or pieces, try simpler problems, or constrain a variable or try a particular case….
Click here for the problem —->
And slowly, they become less afraid to poke at a problem, even if it is completely unfamiliar. Usually, I give them challenging combinatorial stuff, before we have seen it (and importantly, before they have developed or received a formula). But on their last test, they got the following. If you can answer, great. But see if you can make any progress, if you have any productive ideas at all.
A circle and a square have the same area, and share the same center. The radius of the circle is r. Find, in terms of r, the area enclosed by both figures.
JD2718 
This is hard when you can’t draw a diagram…
The total area is the area of the square, plus the area of the 4 pieces of the circle that stick out past the square.
The side of the square is sqrt(pi)*r
To find area of the four little pieces of the circle that stick out past the square you need to look at the “wedge” of the circle they are part of and subtract the area of the triangle that’s inside the circle, and you need to know the central angle of the wedge, a.
Looking at the triangle part of the wedge, a/2= arccos(sqrt(pi)/2).
After that its just calculating — the area of the wedge and triangle, but I don’t think I can do it with this text editor.