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.)
+23252 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.
