Page 29 - Abstract book - TJSSF-2020

Thailand – Japan Student Science Fair 2020 (TJ-SSF 2020)
“Seeding Innovations through Fostering Thailand – Japan Youth Friendship”

Extension of the Pizza Theorem to Any Regular N-polygon

Proof Using Internal 2N-Polygon and External Symmetry

Mai Nozaki
1
Advisors: Hiroki Iwakura , Rick Mabry
1
2

Bunkyo Gakuin University Girls’ Senior High School
1
Louisiana State University
2

Abstract

The pizza theorem states that if a circular pizza is divided into 8, 12, 16 or more slices by
making cuts at equal angles from an arbitrary point, then the sum of the area of alternating slices (every
other slice) is equal. Here, I prove that the pizza theorem can be extended from a circle to determine the
area of any regular N-polygon. The demonstration was performed in four steps.
In the first step, a 2N polygon (hereinafter referred to as "Cutting Polygon") is created inside
the regular N polygon by a certain operation. Then, in the next step, prove the joint definition in the
pizza theorem in Cutting Polygon. In the third step, the area between the regular N-sided polygon and
the 2N-sided polygon (hereinafter referred to as "External Cutting Polygon") is cut by a certain
operation, and a pair of pieces named based on the symmetry used in proof are created. Then, in the
final step, demonstrate the congruence definition in the pizza theorem for a pair of External Cutting
Polygon pieces using "symmetry" and "rotation of the cut line".
As a result, it was proved that the pizza theorem can be extended from a circle to a regular N-
polygon in the case where N is even.

Keywords: The pizza theorem, geometry, N-polygon, symmetry,

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