The Arlington County Board yesterday approved plans for a 13-story office building in Rosslyn, a project that county planners said should complete the planned development of high-rise towers in the area.

"This building is smaller in height . . . than other buildings you have approved since 1970," said community affairs director William L. Hughes as he urged the board's approval of the office building. The tower will be part of a four-building complex on N. Kent Street, overlooking Interstate Rte. 66.

Rosslyn, located across Key Bridge from Georgetown, was a rundown section of pawnshops, lumber yards and storage tanks until the 1960s when the county began an intensive development program. That growth accelerated with the opening in July 1977 of the Rosslyn Metro station, which has given the area direct access to downtown Washington and the Capitol.

County Board member Joseph S. WHoley asked developer Robert Smith about the possibility of erecting a high-rise condominium on the site. "You hear talk about Rosslyn being a ghost town after dark. What about the desirability of the county insisting that this be an apartment or condominium?"

But Smith said that he would have to sell condominium units for $150,000 each in order to recover his building costs.

When Wholey pressed Smith, the developer retorted: "Mr. Whorley, I've been a developer for 25 years. I do this every day and you do it on Saturdays. You just can't do it" financially.

In negotiations with county planners Smith had originally agreed to pay $28,000 for the development of a "vest-pocket" park near the complex. That figure was later upped to $50,000 at Wholey's request.

Two tennants from the adjacent Normandy House Apartments opposed the additional office building. They told the board it would generate more traffice and block their view of Washington.

"We're not asking for any great bonuses or differences in height," said Smith's attorney Barnes Lawson, noting that at least 12 buildings in Rosslyn are taller than Smith's will be.

County officials said that three other Rosslyn highrise office buildings, which will be among the tallest in the metropolitan area, are currently under construction.

"I think it's time to finish Rosslyn and this is going to be the best development we can get on that particular site," said Vice Chairman Ellen M. Bozman who voted to approve the building. She was joined by Wholey, Board Chairman John W. Purdy and member Walter L. Frankland Jr. Board member Dorothy T. Groton voted against it.