When airplane pilots begin a 10 minute wait timer when lightning strikes, is there a geometric reason behind it?
No, because adding candles to candy does not make much sense.
The dimensions are different, so the relative numeric values depend on the choice of units. You could equally state the rule as you have to wait ⅙ of an hour after lighting strike within 5 km (if it's statue miles; 5.5 if nautical) and now the numbers won't match any more.
Besides, it is a 3 mile radius, so the area is about 28 square miles and the numbers never matched to begin with.

The real reason is two-fold:

When there is active storm right over the airfield, somebody could get struck by lightning. Inside aircraft and vehicles is safe, because the lightning will ground through the skin and not get inside, but out on the ramp it wouldn't be. 10 minutes is long enough to consider the storm no longer active.

The storm builds up static charge on the aircraft and vehicles. This is again not a problem for the occupants, because the conductive skin ensure the whole vehicle is at the same potential, but it could cause a static discharge strong enough to hurt someone or set something on fire when touching the vehicle from outside. 10 minutes is considered long enough for these charges to gradually discharge through the wheels to the ground and through the static wicks to the surrounding air.
This would also apply to aircraft that flew near the storm just before landing, but that is normally avoided as storms generate dangerously strong wings.

The 3 mile radius is simply size of typical storm.