Intro: How to Fold a Polygon (featuring Pentagon)
It is always hard to draw a polygon (eg. decagons, nonagons, heptagons...etc) with compass and ruler. The method I discovered is to divide a polygon into smaller triangles. That means you only have to concentrate on one intersected triangle and the relations between the vectors. The other triangles are just repetitions.
Your challenge is how to find out the angle phi for other n-polygons (general knowledge: Trigonometry, Maths - Secondary School). I give you a hint in the following pdf for n-gons.
If you have the solution for phi, don't post it here, they have to solve for their own!
So let's get started how to fold the pentagon.
Step 1: The Crease Pattern
Red: valley folds
Blue: mountain folds
Black: cutting lines
Step 2: Cut, Precrease and Glue
So the next step is to precrease everything and to glue it to a ring. (in the middle there are 6 rectangles, one is for gluieng, overlap the first rectangle with the last one)
Step 3: Fold and Tuck
The next step is to fold and tuck the precreased paper ring into a pentagon. Repeat it for the another side.
Step 4: The Result
And you are finished