1994 AMC 12 Problem 26

Question

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?

5

6

14

20

26

At each vertex of the inner polygon, one

m

-gon angle and two

n

-gon angles fill

360^\circ

Use the regular

k

-gon interior angle

180^\circ(1-\frac{2}{k})

At a vertex of the inner polygon, its interior angle and two angles from the surrounding polygons total

360^\circ.

Thus

\begin{gathered} 180^\circ\left(1-\frac2m\right) \\ {}+2\cdot180^\circ\left(1-\frac2n\right)\\ {}=360^\circ. \end{gathered}

This simplifies to

\frac{1}{m}+\frac{2}{n}=\frac{1}{2}.

With

m=10,

we get

\frac{2}{n}=\frac{2}{5},

so

n=5.

Thus the correct answer is A.

Problem 26 in other years