Dr Ian Plummer
Technical
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 faceon 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
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