Golden Ratio in a Regular Pentagon
The golden ratio, occurs in isosceles triangles measuring 36°, 72°, and 72°. It is the ratio of one of the legs of the triangle to the base. As seen in this applet, the diagonals create a triangle with these measurements if and only if the regular polygon has n=5 sides. If you slide the slider, you will see that different number of sides changes the angle measures so that the sides of the triangle formed by the diagonals no longer has a ratio that is the golden ratio.