He received 869 votes, with his opponent, Graeme Smith, getting 283.
Mr Cadogan, who lives in Clyde, said he was pleased and excited about the trust people had put in him and looked forward to finding out what being a board member was all about.
"It's a big learning curve and, as this is a by-election, everything is already under way [for the new trimester] and I will be working to pick up on things that have already started," Mr Cadogan said.
Mr Cadogan completed his law degree and then worked very much in the public arena, in broadcasting, before he returned to law in Alexandra.