IT COST $94.4 million to build the new addition to the National Gallery of Art. The job is almost done. The angular East Building, designed by I.M. Pei and Partners, won't open until June, but it is already changing the National Gallery's geometery. That once-square institution will never be the same.

The new East Building, first of all, is made of triangles with points. Some are huge, some small. The biggest triangles of all - 150 feet at the base, with sides 225 feet long - is the faceted glass-and-steel space-frame that keeps the weather out but lets the sunlight enter the museum's indoor park. That enormous, complex skylight, with its tetrahedral voids and its pointed panes of filtering, insulating glass, roofs 16,000 square feet, more than one-third of an acre. The airy seven-story-high, wedge-shaped room it covers has already started filling up with art.

thogh the building is still closed to the public and the press, you do not have to go inside to see the graceful mobile, designed by Alexander Calder, that is the new East Building's first work of modern art. Walk downhill from the Capitol towards Pei's knife-shrap marble towers, approach his inset window walls, peer up through the greenish glass toward the skylight high above, and already you can see the red vances of the Calder moving in the air.

Somewhere in the basement, Gallery employees have been at work for weeks tending three computers (for the security system; for temperature, airflow and humidity; and for payroll, purchases and bills). Already guards patrol the new museum's balconies and bridges. Masons are still fitting triangles of marble into the broad stone floor, but their work will soon be finished. Almost any day now, the hardhats will depart, the construction bills will be paid, and the Gallery's trustees, acting for the people, will accept the Mellon gift.

Its funding has three sources. It was paid for by Paul Mellon, the Gallery's president, by his late sister, ailsa Mellon Bruce (once Mrs. David K.E. Bruce), and by the foundation that they formed with moneys left them by their father, Andrew W. Mellon. The old National Gallery building was Andrew Mellon's gift. This city's newest monument, that built by hs children, has already changed the Mall. Soon it will adjust the art - and how we think about the art - that the Gallery displays.

Pei's triangular building is pointed toward the future. The permanent collection that it will house has not yet been assembled, but the new museum will not open empty. Beginning June 1, and for many years thereafter, the East Building will give Washington more international art loans, more international art thought and less emphasis on the Old Masters - those presents from the rich, reflective of their taste - that were once the sole attraction of the National Gallery of Art.

The Gallery once seemed the most old-fashioned, clubby and sedate of American art museums. It was once one institution. It will soon seem three.

The first third of the triad is the Gallery we know, that neo-classical temple that Andrew Mellon paid for and John Russell Pope designed. It is going to be upstaged. the formal skylit galleries of the "West Building," as it is now known, will continue to present a chronological survey of the history of Western painting, but its awkward downstairs spaces have already seen their last big temporary show. Beginning June 1, major exhibits and the crowds they draw will be seen in the new building. The old Gallery will also lose its offices, its library, its buzz. When that happens, the National Gallery of Art - or rather, the West Building - will once again be ruled by contemplative peace.

That half of Pei's East Building that will serve as a museum is the triad's second third. Its new exhibition spaces - 20,000 square feet, below grade, for temporary shows, and 75,000 square feet in the towers and the court - will open full of art, old as well as modern, most of it on loan. Its first exhibit - of suits of armor, jewelled statues, porcelains and paintings, from Dresden in East Germany - is now being designed. There will be no private preview, no invitational dinner. On the morning of June 1, the President will snip a ceremonial ribbon and the people will pour in.

A new Center for Advanced Study in the Visual Arts will occupy the library-and-office half of the East Building. The opening of the center "CASVA" is its nickname is still some years away. The center will be a think tank for scholars from around the world. The 56,500-square-foot, six-story library that will be its core eventually will house 300,000 books, many of them rare, and a photographic archive, probably computerized, of 2.5 millionimages.

In the Gallery's offices today one feels an eerie stirring, a gathering of energy, anxiety, excitement. It is like pulling back a slingshot. The target is in sight at last, the tension is increasing. The Gallery is about to be hurled out of its pst.

"It's like learning a new language," says Gaillard F. Ravenel, the Gallery's chief of installation and design. The diagram before him is full of oblique angles, octagons and hexagons. Ravenel is sitting at a drafting table in what once was the gymnasium of the Gallery's old building, deciding how to put old art in the new. "Actors must feel this way when they are told, 'No more proscenium. From here on in it's going to be theater-in-the-round."

No edifice in Washington is more dignified, luxurious, than the Gallery's West Building. Its benches are cut marble and its elevator doors are covered in soft leather set with studs of brass. John Russell Pope, the architect, provided galleries of remarkable variety, some large enough for tapestries, some small enough for postage stam ps. But its galleries, though varied, all evoke an antique mood. The paintings hang as if installed in aristocratic houses, palaces or temples. Most paintings are rectangular, so are most walls and doorways. In Pope's West Building the right angle rules.

The new East Building, as Ravenel is well aware, is not like that at all.

Its module is an isosceles triangle whose base-to-side proportion is set at two-to-three. The galleries in its towers are hexagonal in plan (imagine two such triangles butted bse-to-base, then wall off their tips and use the pointed rooms left over for utilities and stairs.) Pei's galleries, like Pope's, come in many different sizes - some have 10-foot ceilings, some are two stories tall; some are small, 1,500 square feet, while the flexible open space for temporary shows on the concourse level covers 18,000 square feet - but all of them are governed by unfamiliar angles. None of them is square.

"The old building is static; the new building dynamic," says David Scott, the Gallery's planning consultant who once was the director of the National Collection of Fine Arts. "We're in a different century, the old ground rules are gone. Pope's galleries proceed in stately, ordered sequence, and they are full of traditional reference points - moldings, wainscots, columns - but Pei's galleries have none at all, not even normal doorways. You can enter the hexagonal galleries because one side has been left off. The flat walls of marble slide past walls of glass, the neutral public spaces slide into the galleries. We are learning how to use this dynamic, shifting space without giving up the traditional picture-viewing experience."

Pope's original design set no space aside for temporary shows. The rooms that have been used there for big loan exhibitions have a space so awkward the installers call them "Chinatown." But Ravenel and his staff know that space by heart; for shows in the East Building they will have to start from scratch.

"In the new building," says Ravenel, "the sight lines are completely different. They are not single, they're multiple. If you walk into a rectangular gallery you tend to look at the wall in front of you. Then you turn right or left. In a gallery that is hexagonal, other rules apply. Those old linear axial sight lines that I love so well are no longer there. Because Pei's module is a triangle, and because his spaces flow into one another, one tends to approach objects from many different angles on a multiplicity of paths."

Carter Brown believes that museum installations, even temporary shows, should have a look of permanence. They don't use pgo panels or thin plywood partitions at the National Gallery of Art. For almost every loan show there they construct new spaces with new floor plans and new walls. "When you come here to see Tut, the Alaskan show, the Noh robes, you are not supposed to fell you're visiting the Valley of the Kings, Alaska or Japan." The same standards will be followed in the new East Building. For 18th-century objects in the Dresden show, the Gallery is building temporary rooms whose antique details and finishes conjure up that time.

But other problems face the National Gallery of Art. You can almost hear the capitals when Gallery empolyees talk about "The Move." In a large reference library a book that is incorrectly shelved might as well be lost. How do you pack 100,000 books so that when you unpack them they are still in sequence? Hoe do you plan the still in sequence? How do you plan the stacks so that there is enough, but no too much, shelf space set aside for all the books whose titles begin with K or F? The photographic archive, too, presents its headaches. The challenge is to pack and move a million separate images without misplacing one. "I tend to dread The Movie," says Dorothy Fall, one of the librarians, "but when I feel most anxious, I think about the light. This library is like a cave, you cannot see the weather. Each night when I leave work I feel a burst of pleasure - there is a sun! the sky is blue! - I'm going to like the windows."

When Leonard Jacobson, an associate partner of I.M. Pei's, speaks of the East Building, he does so with nostalgia. "We knwo we're almost done."

The East Building stretches for 405 feet along Pennsylvania Avenue, for 270 feet along 4th Street and for 382 feet along the Mall. Its towers are 107 feet tall. The building seems enormous, but it is only half of the addition to the Gallery. Another structure of comparable size has been built below the ground.

The underground connecting link between the old building and the new one includes a 110-car garage and service dock, some 20,000 square feet of officesand workshops and a 700-seat public cafeteria that has served a million customers since its opening last summer. Building the connecting link was not an easy job.

"We had to slip a new building under an old one," says Jacobson, "and under an existing street. The excavating machines, the moles, had to wend their way through a forest of old piles that held up the old building. They could not be the old building. They could not be bumped. We had to be as careful of all th eutilities underlying 4th street, the 30-inch high-pressure water main, for instance, the telephone cables, a 33,000-volt live electric line. All sorts of stuff. We slipped a new building underneath these lines. The engineering was tricky. The ground is very wet there, and if the building were not waterproof it would tend to leak. If it were not anchored properly, it would tend to float.

"They are in the East Building all sorts of subtle details the visitor will not see. If you look at the walls of the central court, I'll bet you do not see the vents that admit air. Imagine how much water will fall on the 16,000 square feet of the central skylight during a torrential downpour. In the valleys of the skylight, where the tetrahedrons meet, we've built in neoprene gutters. The water will collect there and then run off through holes pierced in the steel space-frame to drains concealed in the walls. We've put in heating wires, too, so we can melt the snow. You won't see any of this. The water will disappear without waterfalls or gurgling. That's not a theory, that's a fact."

"While we were building, construction costs increased some 41 per cent. The budget was not infinite. Here and there we had to squeeze. We were going to put a marble floor in the cafeteria. Instead we used terrazzo made with marble chips. Originally there were no horizontal divisions in the glass walls of the skylight. In order to save money we used smaller panes of glass. In the galleries the metal trim is highly polished stainless steel. In the study center, the equivalent trim is painted aluminum. If Kenneth Clark comes to Washington to write another book, he will do so in an office with standard gypsum-board walls."

Hurley Offenbacher, the Gallery's construction manager, was in charge of building th eSmithsonian Institution's Air and Space Museum when he met Carter Brown. The Air and Space Museum was expected to cost $41.2 million, but it was completed, ahead of schedule, for some $2.2 million less. Brown, who watched its progress closely from his post as chairman of the National Commission of Fine Arts, offered Offenbacher a job.

"The whole East Building is custom-built," said Offenbacher. "the tolerances throughout were about 1/16 of an inch. The skylight was built slowly. It is made of hundreds of odd-shaped panes of glass. It is not supposed to leak. We sent back some 60 panes of glass. We inspected every joint. Then we went up on the roof with high pressure water hoses to make sure that it was watertight. We are having a little trouble with interior condensation, but we think that we can fix that by fine-tuning the flow of air."

"An orderly transition is what we're after," said Charles Parkhust, the Gallery's assistant director."We've been working carefully to make this institution strong enough to bear the future. We've had a department of installation and design since 1971. We've steadily strengthened the curatorial staff, the library, the photo archive. It upsets me that I no longer know the names of all the guards, but now there are too many.

"When I think of the new gallery," Parkhurst added, "I see in my mind's eye an old Thomisitc image - we've been building the earthly pyramid, and we're almost at the apex. Now we have to balance the heavenly pyramid tip-to-tip on top."