Problem:
A regular polygon of m sides is exactly enclosed (no overlaps, no gaps) by m regular polygons of n sides each. (Shown here for m=4,n=8.) If m=10, what is the value of n?
Answer Choices:
A. 5
B. 6
C. 14
D. 20
E. 26
Solution:
The measure of each interior angle of a regular k-gon is 180∘−k360∘​. In this problem, each vertex of the m-gon is surrounded by one angle of the m-gon and two angles of the n-gons. Therefore,
(180∘−m360∘​)+2(180∘−n360∘​)=360∘,
and m=10 gives n=5.
Note. The equation may be written in the form (m−2)(n−4)=8. Its only solutions in positive integers are (m,n)=(3,12),(4,8),(6,6), and (10,5).
OR
Each interior angle of a regular decagon measures (180∘−36∘). The interior angles of the two n-gons at one of its vertices must fill 360∘−(180∘−36∘)= 216∘. The regular polygon each of whose interior angles measures 216∘/2= 108∘ is the pentagon, so n=5.
.jpg)