Because the ring is made out of regluar pentagons, we can work out that each of the interior angles of each pentagon is 108 degrees. By extending the lines that two pentagons share, assuming they will all meet in the middle*, it will create a triangle. As we know each angle of a pentagon is 108** degrees, we know the two base angles of the triangle would equal to 72*** which leaves the top angle to be 36 degrees. As a angle at a point is 360 degrees, and 36 is divisible by 360, it will make a complete ring. Also, as 360 ÷ 36 = 10, we know that the ring will be made out of 10 pentagons.
Pentagons above are regluar pentagons.
*because as the ring is made out of regluar pentagons, by extending the sides, it must make a isoscele triangle. If they do not meet in the middle, it will not creat an isoscele triangle by extending the lines of their neighbouring pentagons.
**as formuale to calculate the the size of a interior angle of a polygon is (n x 180 - 360) ÷ n. (for n being the number of sides that the polygon have.
***because the triangle is made by extending the lines, and angles on a line is 180 degrees, 180 - 108 (a interior angles of a pentagon) = 72.