The five spots at a glance
All figures assume a 2.15 m release, a 3.05 m rim and no air, using the minimum-speed formulas from Module 01. Lateral margin is how far off-line the ball can travel and still clear the rim when centred in depth.
| Spot | Distance | Min-speed angle | Min speed | Lateral margin |
|---|---|---|---|---|
| Free Throw | 4.19 m | 51.1° | 7.13 m/s | ±1.49° |
| Three-Pointer | 7.24 m | 48.5° | 8.97 m/s | ±0.86° |
| Wing (45°) | 7.24 m | 48.5° | 8.97 m/s | ±0.86° |
| Corner | 6.71 m | 48.8° | 8.67 m/s | ±0.93° |
| Half-Court | 13.94 m | 46.8° | 12.08 m/s | ±0.45° |
Why lateral error gets worse with distance
The rim leaves about 10.9 cm of clearance on each side of a centred ball. The allowable direction error is simply that clearance divided by distance:
δ_max = arctan( (D_rim − D_ball) / 2 / x )
= arctan( 0.109 m / x )Depth error can be partly rescued by the rim and glass; sideways error almost never can. From half-court you have less than half a degree — roughly the width of a fingernail at arm's length. In the lab, the Horizontal Direction slider lets you feel this: a single 0.5° step is harmless at the line and fatal from half-court.
Spot-by-spot strategy
Free Throw
The easiest physics on the floor: short, generous laterally, and uncontested. Use it to calibrate. Aim for 52–56°; the extra arc costs almost nothing and gives a healthier 40–47° entry angle.
Three-Pointer (top of the arc)
Needs about 26% more speed than a free throw, so timing errors translate into bigger misses. The lab's perfect zone narrows to roughly ±1% of the meter. Stay close to 48–52°; flatter shots demand more speed and give a smaller entry window.
Wing
Same distance as the top of the arc, but the backboard is now angled toward you. That opens the bank shot: a slightly long shot aimed at the back-right of the rim can kiss the glass and drop. See Module 04 for the reflection maths.
Corner
The shortest three on an NBA-style court, 0.53 m closer than the top. But the backboard is almost edge-on, so there is effectively no glass to help — misses long tend to fly straight through the back. Precise depth control matters more here than anywhere else.
Half-Court
A different sport. Required speed exceeds 12 m/s, drag becomes a major term and both the timing window (≈ ±9 ms) and lateral margin (±0.45°) are tiny. Aim slightly flatter, near 46–47°, where required speed is lowest, and treat any make as a bonus.
Expected value: which shot is worth it?
The core equation of shot selection is simply points multiplied by probability:
EV = points × P(make) Break-even: 3 × p₃ = 2 × p₂ → p₃ = ⅔ · p₂ A 36% three (1.08 pts) beats a 52% two (1.04 pts)
Use your shot chart to calculate real EV per spot. If your corner three runs at 40% (1.20 points per attempt) while a long two sits at 45% (0.90), the data is telling you where to stand — just remember to wait for a large enough sample, as explained in the Analytics Lab.