Prince William County school officials, struggling for years to keep up with the rapid growth of the county's student population, announced last week that at least four new schools will be needed by 1990 at an estimated cost of $34 million.

With more than 36,800 students, the Prince William system is the third largest in the state.

Officials predict that if population growth in the increasingly suburban county continues at its current rate, student population will reach more than 41,000 by 1990.

According to the report, three new elementary schools and at least one new high school should be operating by that time.

The elementary schools would be in the Lake Ridge area, in western Dale City and at a site west of I-66. A high school site has not been designated. A fourth elementary school may be needed in the western end of the county if current growth trends continue there, a report said, which would bring the cost to $40 million.

Officials are reluctant to discuss how the schools would be funded. According to School Board Chairman Gerard Cleary, there are no plans to approach the Board of Supervisors about it in the near future.

"Traditionally, we've built schools on a pay-as-you-go basis," Cleary said. "That's why our taxes are so high here."

Last year, under pressure from the supervisors, the School Board decided against a seventh high school that it hoped would be part of a bond referendum slated for next spring.

As a temporary measure to relieve elementary school crowding where the problem is greatest, the report recommends boundary shifts for Coles and Sinclair elementary schools. Students living in some neighborhoods near Sinclair may be transferred to Loch Lomond in western Prince William.

Students living near Coles would be moved to the Bennett School in Manassas. Two public hearings on the boundary changes were held this week, one at Coles and one at Sinclair.

In addition, officials said they may buy more trailers to be used as temporary classrooms.

According to Cleary, while midcounty schools and population were not considered in the report, people are settling there in increasing numbers. "We're really concerned about that, and we're watching that growth pattern very carefully," he said.

One of the problems the school district faces is a lack of construc- tion sites in the western end of Prince William. Historically, school sites are one of the offers made voluntarily by developers in exchange for certain considerations on their projects.

The county requires 15 acres for an elementary school, 30 for a middle school and 70 for a high school.

The report made no recommendation regarding middle schools pending a study of what school spokeswoman Kristy Larson called "the middle school philosophy." She said middle schools have been seen as "transitional," a term she tagged "obvious -- and vague."

Larson declined to say whether the study will determine if sixth graders should be moved to the middle school with seventh and eighth graders.

The report said that the county's nine middle schools will be able to handle projected growth at least until 1990.