Tub vent

My toilet and bathroom sink are right next to each other and both are vented with a 2" pipe, with the separate run for the sink that connects about 4 feet up as I've seen in the diagrams for venting.

I can also run a vent for the bathtub but it will be a pain as it's across the room (about 5 ft).

Is it really necessary to have a vent for the tub or does the overflow drain let it drain well?

THANKS.

It needs a vent after the trap. The overflow won't protect the trap seal.

The overflow is NOT a vent, and HOW the tub connects to the main line determines whether it needs a separate vent or not.

Yes the vent will be about 5 feet from the tub after the p trap. I'm putting the vent right between the toilet and sink in the wall (2" pipe).

Toilet > vent < Sink The sink and tub will wye into the main 3" drain coming out of the toilet and I will also run a 2" pipe from the sink upwards into the vent.

Like that deep space Hubble photo -- I have it framed in my home.