Recently a friend asked why I haven’t put up any new POWs (Problems of the Week). Part of the reason is because I got lazy but the other part is because the show Numb3rs is what inspired me before and I haven’t been watching any reruns lately . Yesterday I started watching a new show called Scorpion which has a little math in it and the second episode got me thinking about inscribed shapes. Take the following three shapes for example:

Above is a picture of a circle inscribed in a square which itself is inscribed in another circle. Now if I make the inner circle red and the remaining area black the image ends up looking as follows:

Using the information given, is the red area bigger or the black area? If you remember a few things from trigonometry/geometry I’m sure this math problem will be a breeze . As usual the answer to this problem of the week became available a week after the POW was published and can be found here.

If the objects are 2 dimensional, the black area is bigger. But there is not enough information to state otherwise. If the red area is 3 dimensional and the black area is 2 dimensional, the size of the red would be greater.

Yes, the shapes are two dimensional circles. Additionally this is not meant to be a trick question but an application of what we learned in geometry/trigonometry. The images are more or less a way to visualize the problem.

However, I have a second thought. The wireframe drawing could imply that the black circle is a 3 dimensional object, and the square area represents its hollow inside. In which case, if the red circle a sphere, the red would be bigger.

It did sound like a trick question. Never mind.