Question What is the regulation height and the regulation distance for a dart board
Answer The dart board is hung so that the centre of the bullseye is 1.74m (5'8") from the floor.
The throw line, which is also referred to as the oche (pronounced "ockey")is generally located 2.37m (7'9.25") from the face of the dartboard measured horizontally. This is the recognized world standard asset by the World Darts Federation and is played as such in most areas.
There are variations in distance and some people play from 2.29m (7'6"), 2.44m (8') and as far as 2.74m (9'), In Beeston in Nottingham UK I played in a 6ft league!.
Since most dart boards are about 11/2" thick it is important to measure to the oche from the front of the board and not the wall. By far the easiest way to do this is to pin some string to the centre of the bull then measuer diagonally to the floor 2.931m (115-3/8"). This will reduce the margin of error as long as the bullseye has been placed at the correct height and the floor and the wall make a right angle.
This answer contains more than you asked for but never harms to make sure its all covered.