So I ran a gladiator, martial class only dnd one shot where my players fight other fighters from other planes. to make a long story short, a monster I created teleported behind a polearm master player character to try and do a sneak attack. The player argued that he could do an opportunity attack because the monster entered his pikes range even though his character could not see the monster. We argued back and forth and I eventually conceded to him and let his character make an opportunity attack since he argued the area of effect for polearm master is just whatever his spear range is. 

My logic, however, is that the area of effect is only what his eyes can see. Meaning if you have polearm master and your pikes range is 15 ft the area of effect is more like a cone limited to the degrees of sight that the character has. 

Effects like this are in the full 360 degrees - the game presumes that, in the middle of a battle, you are keeping your eyes open and aware of all your surroundings. 

A DM could homerule otherwise,  but that would be a homerule,  of rules as written.

what would you rule as a DM if the player character with polearm master has his view obstructed? like if they were in a cloud of smoke or if the enemy is invisible. If an enemy moves within their polearm range to attack them? would they get the opportunity? I'm just curious cause the wording for polearm master is vague

*You can make an opportunity attack when a hostile creature **that you can see** moves out of your reach.*

RAW, You need to be able to see the creature to take opp. Attacks, so invisible creatures or creatures on heavily obscured areas don't provoke opp. Attacks.

But anything approaching from the 360o POV of the character that is visible, would.

Yeah, one would assume a martial fighter would be trained to maintain a 360 angle in a fight. Although different things may effect that. If it's 1v1 and the creature isn't invisible or somehow heavily obscured (or invisible) I would rule the PC sees them 100%.  If it's a fight with multiple people (both ally and enemy or just multiple enemies) or the enemy is partly obscured, then I would probably do a stealth vs perception check to see if the creature took advantage of a moment the PC focused somewhere else.

Quote from Caerwyn_Glyndwr >>

Here is what Rules as Written say about attacking an obscured target: 

Combatants often try to escape their foes' notice by hiding, casting the invisibility spell, or lurking in darkness.

When you attack a target that you can't see, you have disadvantage on the attack roll. This is true whether you're guessing the target's location or you're targeting a creature you can hear but not see. If the target isn't in the location you targeted, you automatically miss, but the DM typically just says that the attack missed, not whether you guessed the target's location correctly.

When a creature can't see you, you have advantage on attack rolls against it. If you are hidden--both unseen and unheard--when you make an attack, you give away your location when the attack hits or misses.

Strictly speaking, you can get an opportunity attack against an invisible/obscured creature that is moving since you can judge where they are based on sound--the attack is just at disadvantage since you are swinging your sword blindly in the direction of something you can hear but not see.  

Again, this is something where you could do some homeruling to improve--for example taking passive perception into account to see whether they can even detect the movement or movement direction in the first place.  

This is Straight out of the PHB I bolded the appropriate sentence 

In a fight, everyone is constantly watching for a chance to strike an enemy who is fleeing or passing by. Such a strike is called an opportunity attack.

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

You can avoid provoking an opportunity attack by taking the Disengage action. You also don’t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction. For example, you don’t provoke an opportunity attack if an explosion hurls you out of a foe’s reach or if gravity causes you to fall past an enemy. 

In the case of the OP the player is error the creature is neither fleeing or passing by the creature teleported or was teleported behind the gladiator and is covered by the rule. 

Quote from Xeresia >>

Quote from Caerwyn_Glyndwr >>

Here is what Rules as Written say about attacking an obscured target: 

Combatants often try to escape their foes' notice by hiding, casting the invisibility spell, or lurking in darkness.

When you attack a target that you can't see, you have disadvantage on the attack roll. This is true whether you're guessing the target's location or you're targeting a creature you can hear but not see. If the target isn't in the location you targeted, you automatically miss, but the DM typically just says that the attack missed, not whether you guessed the target's location correctly.

When a creature can't see you, you have advantage on attack rolls against it. If you are hidden--both unseen and unheard--when you make an attack, you give away your location when the attack hits or misses.

Strictly speaking, you can get an opportunity attack against an invisible/obscured creature that is moving since you can judge where they are based on sound--the attack is just at disadvantage since you are swinging your sword blindly in the direction of something you can hear but not see.  

Again, this is something where you could do some homeruling to improve--for example taking passive perception into account to see whether they can even detect the movement or movement direction in the first place.  

This is Straight out of the PHB I bolded the appropriate sentence 

In a fight, everyone is constantly watching for a chance to strike an enemy who is fleeing or passing by. Such a strike is called an opportunity attack.

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

You can avoid provoking an opportunity attack by taking the Disengage action. You also don’t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction. For example, you don’t provoke an opportunity attack if an explosion hurls you out of a foe’s reach or if gravity causes you to fall past an enemy. 

In the case of the OP the player is error the creature is neither fleeing or passing by the creature teleported or was teleported behind the gladiator and is covered by the rule. 

 The PC had Polearm Master.

• While you are wielding a glaive, halberd, pike, quarterstaff, or spear, other creatures provoke an opportunity attack from you when they enter the reach you have with that weapon.

Im pretty sure this is a case where the specific overrules the general. If they see you, (which we've already argued about if they could or not) and you enter their range, they get an Opportunity Attack.

As mentioned characters have 360 degree view so there is no facing in 5E. Also teleportation doesn’t trigger Opportunity Attacks (see spoiler)

And if the atracker is obscured, like you mentioned in fog or invisible, then use the unseen attacker rules. 

There is an optional rule for Facing in the DMG (https://www.dndbeyond.com/sources/dmg/running-the-game#OptionalRuleFacing).

But standardly the game assumes you are constantly looking in all directions for danger during a fight.