The Effect of Hoop Upright Diameter on the Acceptance Angle of a Hoop

To save others scratching their heads on how to calculate the acceptance angle of a hoop the detail is given below. The answer is of current interest where it is planned that 3/4” diameter hoop uprights be used instead of the ‘normal’ 5/8” ones in an International competition.

The above diagram is the key for calculating the angle theta, Φ, which is the maximum angular deviation from playing face-on to the hoop where the ball will just pass through the hoop without touching the uprights. 

The following variables are used:

r(hoop)	Radius of the hoop upright	5/16” or 3/8"
ball	Diameter  of ball	3 5/8” = 29/8”
r(ball)	Radius of ball	29/16”
gap	Clearance between the ball and the hoop uprights	Taken as 1/32”

The blue triangle is the route to the answer.  From the diagram:

the hypotenuse = r(hoop) + r(ball) + ½ gap
the adjacent side = r(hoop)  + r(ball)

Standard relationships give cos Φ = adjacent / hypotenuse, hence

cos Φ = [ r(hoop)  + r(ball) ] / [ r(hoop) + r(ball) + ½ gap ]

For a 5/8” diameter upright theta = ± 6.927 degrees, 13.854 degree opening

For a 3/4” diameter upright theta = ± 6.828 degrees,  13.656 degree opening