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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.)

 Feb 15, 2020
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There are 360o around a point.

 

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

 

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

Each interior angle of a regular triangle is  60o

 

Subtracting:  360o - 135o - 60o  =  165o

 

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

 

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

                           --->            (n - 2)·180o  =  165o·n

 

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

 Feb 15, 2020

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