Real Mathematics – Geometry #16

Smell of a Cake

I smell something wonderful. I beam myself up to the kitchen to investigate the source of this smell. I find it: My mother’s chocolate-chipped cake. As I leaned towards the cake, someone grabs my arm: Mom caught me…

I use the emotional card. She doesn’t buy it anymore. My opponent is experienced; my opponent is winning the battle!

As I was thinking of giving up, she offers me a deal. If I can cut three equal pieces out of this cake, one of the pieces will be mine.

Mom’s conditions:

• Only instrument of measurement allowed for the cut is a compass.
• Goal is to cut three pieces that have the same area. Size of the pieces is up to my cutting skills.
• While making the cut, small differences (as if one area is 3,04 and other is 3,09) will be ignored by the mother.
• Most important condition: Pieces must be in the shape of a ring.
• You only have one chance for cutting. There is no turning back after the knife touches the cake.

Art of Cutting Cakes

I tried to find a method on paper because I satisfy all the conditions for the cut.

First I drew a circle that has center O:

Then I created a chord which is as long as the radius of the circle:

I placed the chord on random places inside the circle and marked chord’s midpoints:

I chose any of those marked points and drew a new circle that has center O:

Then I followed the same procedure inside the new circle:

And finally I did the same things for the third time:

Areas which I colored with pens are equal to each other:

For those who wonder the areas, you could calculate and see the approximate results.

One wonders…

I found out that it is possible to cut equal areas that are ring-shaped with using only a compass as mother asks.

Now think: How long the chord should be in order to cut the biggest possible piece?

M. Serkan Kalaycıoğlu