We know Cleopatra and her friends were big on hair removal, not only for cultural and religious reasons but to smooth the surface for body painting.

Many American women today -- particularly during bathing-suit season -- prefer that exposed portions of their body (sans head) be hair-free. (This does not necessarily go for Europeans, however, nor for women who reject as unnatural the business of hair removal.)

All legends to the contrary, hair removal, including shaving, does not make the renewed growth thicker, stiffer or quicken regrowth. "The hair root, which determines the structure of hair, is located below the skin surface and is not affected by anything done to the dead hair shaft at or above the surface," says the American Medical Association's "Book of Skin and Hair Care."

The Food and Drug Administration, which recently took a close look at hair-removal products, reports that "Most of these methods (of hair removal) are safe, if used according to directions and with the necessary expertise. But there is no risk-free method of hari removal, and consumers should be aware that some such products are often the subject of exaggerated advertising."

A rundown on methods:

Shaving: The most popular method, but one that needs to be repeated most frequently. Wet hair is easier to shave than dry; soap or shaving cream make shaving more comfortable. Price can be as minimal as a new blade.

Simple Tweezing: Imparactical (though some try) for arms and legs, and briefly painful. More long-lasting than shaving, usually up to six weeks, but not usually permanent.

Abrasion: Perhaps the oldest hair-removal method. Friction from pumice stone, fine sandpaper or other abrasive surfaces wears hair off the skin. Less lasting than tweezing.

Bleaching: Usually not harmful to the skin, but should be tested on a small area. Duration varies with individual. Continued use may make the hair fragile.

Depilatories: Cream or aerosol applied to skin surface. (Accountable for almost $20 million in 1978 drugstore sales.) FDA warns it should not be applied on irritated or open skin, not used after or before applying antiperspirants. Should be patch-tested first. Price varies with product.

Waxing: Far more popular in Europe than here (Guerlain in Paris uses 5 tons of wax yearly). Heated wax is applied to the skin, then stripped off, removing hair much as tweezing, but more quickly, over a large area. Can be done at home with commercial preparations of heated or cold wax.

Visible growth necessary for waxing to be efficient. Usually lasts six weeks. Price varies with amount of area waxed. Pain eased by temporary numbing of skin with ice.

The ancient Lebanese method combining sugar and heated lemon juice and working into a taffy-like ball is still used some places.

Electrolysis: Hair root is destroyed with an electric current emitted through a very fine wire inserted into the opening of hair follicle. Hair is then removed from follicle by tweezers. Requires treatment by approved professionals in many regions, including Washington and Maryland. (In Virginia, there is no state law requiring a license to practice electrolysis.)

Considered the most permanent.But some complain it is painful, expensive, leaves scars, and regrowth occurs, while others are enthusiastic about results. According to electrolocist Sheri Kaye, who does much "tummy and pubic-hair treatment during bathing-suit season," revised insulated needle and ice applicaton reduce pain. Average charge: $10-$12 for 15-minute session.

Depilatron: Hair is grasped above the skn by operators using an electrically charged tweezer, a method developed as an alternative to electrolysis "without the pain," claims Michael Lipschutz, treasurer of the company. The skin is not punctured and the hair acts as a dielectric (the same idea as radio frequency), he says. According to spokesman William Rados, FDA considers Depilatron "as effective as regular tweezers," and "has never seen any evidence that the electronic process contributes anything. The claim for permanence makes the labeling false."

Recent depilatron ads no longer claim permanence and its brochure states: "Permanent hair removal is almost never achieved in a single treatment . . . Permanent results may not always be achieved."

Electronic Tweezers: Aside from being awkward and difficult for personal use, the FDA says these, too, are no more valid than regular tweezers.

As Judith Willis concludes in FDA Consumer, (October, 1979)," . . . what's hair today may not be gone tomorrow."