Basketball Geometry: Because I’m no good at playing basketball

Editor’s note: Today is Matthias’ birthday, and as such it appears that his maternal instincts have kicked in, and his biological clock is ticking, so today he’s writing for the kids. This is a very good post, though not generally what you may find at North and South of Royal Brougham. However, you may eventually have to get used to finding atypical posts here, as our post type will eventually evolve. This post is very good, and very Matthias. Happy Birthday big guy.

I used to go to Cascade Basketball Camp every year here in Oregon, and every year Barry Adams reminded us of the 30, 50, 70 rule. A math teacher, perhaps at Glencoe or South Salem, had made a few calculations…

A ball that enters the rim at a 30-degree angle “sees” only nine inches of rim, we learned. It turns out, the ball is only nine inches in diameter, so it’s harder to fit it through the hoop cleanly at that angle. A ball nearing the rim at a 50-degree angle has access to 13.8 inches, and finally at a 70-degree angle, the ball sees nearly 17 inches of rim. Ignoring a few fussy details concerning the constantly changing angle of a ball in flight,* these calculations are correct and fairly simple to derive.

The lower the angle of approach, the less the ball’s rim availability will be (the blue line, sort of). The expression for rim availability is simply 18sin(θ).

The lesson is not meant to send kids back to the dorms to pull out their TI-83’s. It’s meant to encourage kids to get a little more arc on their shots. On the basketball court, of course, there is no time for rigorous math. But now, after years of playing and with only a part-time job, I have all day to answer perhaps a more pertinent question.

At what angle should the ball be shot to achieve a certain angle of entry?

At first one might think that the angle at which the ball enters the hoop is equivalent to the angle at which it leaves the shooter’s hand. However, that is only true if the shooter releases the ball at exactly ten feet—the height of the rim. Even a tall, athletic player taking a jumper doesn’t likely release from the height of the rim. For 99.9% of all shooters, the angle of the shot must be greater than the angle of the desired entry.

A three-point shooter that released the ball from a height of seven feet at a 45-degree angle could pause his film session, pull out his protractor, and measure the ball’s angle of entry into the hoop at just 35 degrees. Had he instead fired a 60-degree rainbow, a shot on target would enter the hoop at 55 degrees. Somewhere in between is probably the right compromise between maximum rim availability and feasible accuracy.

So the next logical question is this. What if he moves in or moves back? Does the distance affect the angle of entry? You bet it does. Check out the diagram.

Both players are releasing the ball at a 45-degree angle from the same height. However, the shot from 10 feet away enters the rim at an angle of just 22 degrees, while the longer shot goes through the hoop at 35 degrees (as I mentioned earlier). The diagram was drawn using the physics equations for two-dimensional projectile motion, and the diagrams are all scaled appropriately, so there are no illusions here. You can see the massive difference in the angle of entry from the two different positions on the court.

Most basketball players pick these things up intuitively. If they don’t, then they probably don’t shoot well, and they probably are no longer basketball players—or they’re posts. But it’s worth reminding younger players of the importance of appropriate arc on their shots, and it’s okay to tolerate reduced arc for three-point shooters in exchange for some accuracy.

 

*A slightly more accurate formula would account for the ball clearing the front of the rim, and then having a few split seconds to dive even more steeply before reaching the back of the rim.