There are only regular polygons being used, so we can assume that the interior angle of every pentagon used in the diagram being 108. We can confirm this by using the equation ((n-2) x 180) / n , where n is 5 to get 108.
Where two regular pentagons connect, the sum of the two interior angles, and x, is equal to 360. When solving the equation 2(108) + x = 360, we find x to be equal to 144.
In the unknown polygon in the center, 144 would be the interior angle. We should find the exterior angle to find the amount of sides. We do this by taking 144 away from 180 to get 36.
When we divide 360 by 36, we get 10, which is the number of sides the polygon in the center has, and as such, is the number of pentagons needed to make a ring.
The final equation for how to make a ring out of any shape is attached as a photo below.