Peter has a set of polygonal tiles, where all the polygons are regular and have the same side length. He find that two pentagons and a decagon can fit together perfectly, as shown below.

https://latex.artofproblemsolving.com/c/c/c/cccb8f947bb1012a76ea430ee7e84153d95c73f0.png

Peter also find that a triangle, an octagon, and an n-gon also fit together perfectly. Find n. (Remember that all the tiles are regular polygons.)

Guest Feb 15, 2020

#1**+1 **

There are 360^{o} around a point.

The formula to find the number of degrees in each interior angle of a regular polygon is: (n - 2)·180^{o}/n

Therefore each interior angle of a regular octagon is: (8 - 2)·180^{o}/8 = 135^{o}

Each interior angle of a regular triangle is 60^{o}

Subtracting: 360^{o} - 135^{o} - 60^{o} = 165^{o}

This means that each interior angle of the regular n-gon is 165^{o}

Using the above formula: (n - 2)·180^{o}/n = 165^{o}

---> (n - 2)·180^{o} = 165^{o}·n

If you solve this equation, you will get the number of sides of the unknown regular polygon.

geno3141 Feb 15, 2020