Regular polygon to triangles

Number Theory Level 3

Choose 3 sides of a regular $n$ -gon such that no 2 of the 3 are consecutive. Then extend the chosen sides in both directions until they intersect one another and form a triangle.

In the diagram, for example, around the purple nonagon with $n=9,$ an equilateral triangle has been formed by choosing 3 equally apart sides, which satisfies the inconsecutiveness requirement.

We can see that the other triangle formed—an obtuse triangle—also satisfies the same requirement. So, this picture shows that two unique triangles meeting the requirement can be constructed from a regular $9$ -gon.

How many unique triangles can be formed from a regular $360$ -gon?

The answer is 2700.

1 solution

Jeremy Galvagni
Sep 1, 2018

Basically, you need to partition $n-3$ into three parts, each less than $\frac{n-2}{2}$ . Which is almost a sequence in OEIS

In that sequence, $a(m)=round(\frac{(m-3)^{2}}{12})$ but if $n$ is even, $m=\frac{n}{2}-3$

So our formula for even $n$ becomes $round(\frac{n^{2}}{48})$

The answer is then $\frac{360^{2}}{48}=\boxed{2700}$

Regular polygon to triangles

The answer is 2700.

1 solution

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