I know, I know… Cutting Pizza (part 1) sounds a little of for mathematics, but I assure you it is way more interesting than you think it is. And firstly I have to say thank you to the person that got me into this topic recently. For this post I wanted to write some things I have found interesting when searching for the answer at the following question:
Out of the Box Thinking: What is the easiest way to cut a pizza into 11 equal slices?
One of the first things that comes up immediately is the image of the circle split into 11 parts, like this:
But something inside me was screaming out loud when I saw the image… something made me feel strange. And in a couple of seconds I realized the problem: this solution gives the impression that 360 (degrees) can be divided by 11, but 360 cannot be divided exactly by 11. I took my calculator out and it gets something like 32.7272727272….bla, bla, bla… Moreover, if you think at the length of the circumference or the area of the circle, we get into this problem with 2πr or πr*r, which cannot be divided exactly by 11 unless r is a multiple of 11, but I am sure not all the pizzas out there have radiuses divisible by 11. So I thought that this image is just an approximation which could be used in a restaurant and it is ok, but I wanted to find some other method, something that worked for any radius.
My favorite way to solve this question is the following interesting method:
First, mark out the diameter (you can do this in a variety of ways, e.g. finding the center via perpendicular bisector of two chords). Next, divide the diameter into 11 equal segments, by similar triangles. Call the points on the diameter D0,D1,⋯,D11 in order. Next, cut out 10 semicircles with diameters D0D1,D0D2,⋯,D0D10 and 10 semicircles with diameters D11D1,D11D2,⋯,D11D10. This gives you 11 blade-like pieces that have equal areas.
And the image looks incredibly beautiful if you ask me and it goes well with my mathematician heart:
There is a beautiful explanation using area of a circle which proves the result above. Also this method doesn’t touch the problem of 360 degrees and you don’t need a special radius in order to get the right answer. It goes deeper into the area of the circle and I think it is more appropriate to the question. This could transform into something really interesting for an activity in a lesson on the area of a circle. The question is there, I just have to come up with the rest, but I would love to give it a try.
This all cutting the pizza story is a well known problem for those that study geometry, circular geometry to be more exact. I did not know much about this and I have been researching a little lately about it. And everything starts with the beautiful circle.
Hope this made you a little more curious in this subject. Stay tuned because I will write more about this. And because a post about pizza without an image of a pizza is nothing… Bonn Appetite!
Don’t forget that maths is everywhere! Enjoy! ~LThMath~