Pizza theorem

8 parts (n = 0):
green area (odd) = orange area (even)

12 parts (n = 1):
green area (odd) = orange area (even)

The Pizza Theorem is a mathematical theorem in plane geometry . It describes an equality of area that arises when the circle is broken down into partial areas.

If you lay straight lines ( ) through any inner point of a circle , so that two adjacent straight lines each intersect at an angle of , then you get a division of the circle into areas . If these are numbered clockwise, the sum of the areas with even numbers is equal to the sum of the areas with odd numbers. ${\displaystyle 4+2n}$ ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle {\frac {360^{\circ }}{2\cdot (4+2n)}}}$ ${\displaystyle 2\cdot (4+2n)}$

The name of the theorem comes from the cutting technique used to cut a round pizza into pieces. If you place the knife at any other point on the pizza instead of at the center, you have exactly the above situation.

literature

• Wolfgang Kroll, Joachim Jäger: The Pizza Theorem . (PDF) In: Mathematica Didactica , Volume 33, 2010
• Rick Mabry, Paul Deiermann: Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results . (PDF; 185 kB) In: The American Mathematical Monthly , Volume 116, No. 5 (May, 2009), pp. 423–438 (English) ( JSTOR 40391118 )
• Larry Carter, Stan Wagon: Proof without Words: Fair Allocation of a Pizza . In: Mathematics Magazine , Vol. 67, No. 4 (Oct. 1994), p. 267

Web links

Commons : Pizza Theorem  - collection of images, videos and audio files

• Eric W. Weisstein : Pizza Theorem . In: MathWorld (English).
• Sourcebook for pizza theorem (English)