## Concentric Circles Area Puzzle

This morning I saw this post from Ed Southall (@solvemymaths):

And thought, that looks an interesting puzzle. I’ll have a little go. I think you should too, before reading any further…

Ok, so this is how I approached it. First I drew a sketch:

I assigned the arbitrary variables r and x to the radii of the larger and smaller circles respectively and used the fact that tangents are perpendicular to right angles, and the symmetry of isosceles triangles, to construct two right angled triangles.

I wrote an expression for the required area in r and x. Used Pythagoras’s Theorem to find an expression for x in terms or r, subbed it in and got the lovely answer of 25pi.

*An interesting little puzzle, did you solve it the same way? I’d love to hear alternative solutions.*

I tried this with my Yr9 class with mixed results. I blogged about it here:

https://mhorley.wordpress.com/2015/06/21/geometry-problems/

I’ll definitely try it again with another class but not try counting the squares next time!

Love it, a great investigation there.

I love this problem! Such a surprise invariant. Made a GGB visualization, if that’s of interest. http://tube.geogebra.org/m/224259

Certainly of interest! Thanks.

Assuming there is a unique answer, then reduce the radius of the inner circle until it vanishes, then you just have one circle with diameter 10 cm, so area 25pi sq cm.

Now, this may not be a satisfying solution, but:

(a) for a student in math class or a math competition, making that starting assumption is contextually reasonable, if not mathematically reasonable

(b) this could be the start of a more full investigation where the student figures out the answer (using this assumption) and then checks to see if that is correct for the general set-up.

For a Maths Challenge or US SAT (multiple choice), this is certainly a very appropriate way to solve this problem, but otherwise the assumption is very strong and should be confirmed.

I’ve been teaching UK students to do the US SAT and they find it very hard to make this generalising leap: after it’s so strongly discouraged at GCSE and A-level.