Well we had a lot (maybe not quite as many as 37) at DS2's 6th birthday, & that was in the house, & it's not a particularly big house.
Back room was cleared of furniture, so we had all games there, party tea was a picnic on the floor, & I had 2 DDs aged 14 & 17 helping as well as me & DH. We survived. (Recommend lots of games of Dead Lions )
remember musical stand ups is better than musical bumps! (stand up when music stops!) Also do 2 smaller pass parcels and 2 circles as they wont keep attention for long! Have a colouring table available for the kids who are out of games early, maybe they can make placemats to eat food off of or decorate their own paper plates?