The Arlington County Board yesterday approved the first highrise office building to be constructed in Pentagon City, the last major undeveloped tract of land in the county.

Located south of Shirley Highway (Interstate Rte. 395) and the Pentagon, the controversial 118-acre development is scheduled to be built during the next decade at an estimated cost of $600 million.

The project, almost three times the size of Rosslyn, will contain a town house development, shopping center, hotel complex and numerous office buildings. Arlington officials say they expect Pentagon City to be the most profitable development in the county when completed.

Like the Rosslyn project, Pentagon City has been embroiled in controversy and has been the subject of a two-year court battle. However, opponents of the project who claim it will cause dangerous levels of noise and air pollution, yesterday won only the backing of board member John W. Purdy as the board voted 4 to 1 to approve the new building.

Construction on a small portion of Pentagon City by a developer, Rose Associates of New York, began in 1977, but yesterday's decision will allow the first office buildings in the area.

"This building is strategically located adjacent to the Metro station entrance," said deputy county planning director Thomas C. Parker who urged approval of the 12-story office structure. The building will be located at the intersection of S. Hayes and S. 12th streets.

"This building is indeed attractive," agreed John H. Quinn Jr., "but, alas, this proposal is part of a massive operation that concerns and angers a substantial part of the citizenry. This is a massive high-rise, high-density development."

Quinn headed a group of South Arlington residents who two yeards ago unsuccessfully sued the board over its earlier rezoning of the Pentagon City tract. That suit was turned aside by the U.S. Supreme Court last year.

"The air pollution problems at Pentagon City will be tremendous and impose great health problems," said Derk Swain, who asked the board to impose stringent pollution requirements on the developer. The board, by a 3-to-2 vote, rejected Swain's request.

Rosslyn, the county's other controversial development, was the subject of a closed-door briefing by County Attorney Jerry K. Emrich. He told the board about his reply to a federal suit filed last month against the board and developers of five high-rise buildings planned for Rosslyn.

That suit, initiated at the request of Interior Secretary Cecil D. Andrus, seeks to halt work on the buildings, which Andrus has called "monsters." Federal officials have said that the buildings, one of which is to be compared this month, will ruin the carefully planned Washington skyline. All of the buildings will be considerably taller than those now in Rosslyn.

In court papers filed Friday, Arlington disputes the federal claim that the buildings will violate the "visual integrity of our nation's capital" According to court papers, Arlington contends that "the buildings in question will improve the quality of civilization available to persons in the urban area of and around Washington."

The board also denied allegations of impropriety in approving the Rosslyn buildings, challenged the jurisdiction of the federal government in a local land-use matter and charged that federal officials had "delayed unreasonably" making known their objections to the buildings.

Last week U.S. District Judge Oren R. Lewis refused to grant a temporary injunction stopping work on one of the buildings, which has been under construction for more than a year. The case against the developers and the board is scheduled for trial on Jan. 22.