Page 18 - Abstract book - TJSSF-2020

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

Finding a relationship between the number of vertices of a regular polygon
inscribed in a circle and the number of regions made
By its perimeter and diagonals

Supakorn Thepchan , Samonrach Sutthiboon ,
1
1
Advisors: Maitree Somboon , Detchat Samart
1
2

Princess Chulabhorn Science High School Chonburi
1
Department of Mathematics, Burapha University
2

Abstract

In 1998, B. Poonen and M. Rubinstein established a correct relationship between the number of vertices
of a regular polygon inscribed in a circle and the number of regions formed by its perimeter and
diagonals using advanced techniques from algebra and geometry. The aim of this research is to explain
this relationship using an easy approach. In our study, we use GSP program to illustrate and count the
number of regions made by the perimeter and diagonals of a regular polygon. Based on our observation,
the relationship between the number of regions and the number of vertices, say n, of a polygon can be
separated into 2 main cases depending on parity of n. When n is odd, we find that the number of regions
is n4-6n3+23n2-18n+2424, which agrees with the result of Poonen and Rubinstein. When n is even, we
consider the two subcases, depending on whether n is a multiple of 4. If n is a multiple of 4, we
hypothesized that the number of regions is 9n3-72n2+176n 16. If n is not a multiple of 4 (except n=2),
the number of regions becomes 9n3-72n2+188n16. However, the last two formulas are valid only for
even number n from 4 to 14. To obtain a correct formula for larger even number n, one might need to
consider more possible cases.

Keywords: vertices, a regular polygon, diagonals, circle, the number of regions

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