Cameron Diaz is happy to admit she’s an accidental actress. Fifteen years ago, as a 21-year-old model with no acting experience, she auditioned for her first movie, The Mask, pretty much on a lark. Cut to Hollywood, 2009. Cameron Diaz is the top-earning actress in the world, regularly banking $15m a movie. According to Forbes magazine, she made more than $50m last year. That’s a hell of an accident.

You don’t, however, remain one of Hollywood’s biggest stars for more than a decade by accident. Diaz has done fabulously well in films as diverse as The Sweetest Thing, Charlie’s Angels, Being John Malkovich and Shrek, where she is the voice of Fiona, by trading on her innate winning appeal. As the 36-year-old actress settles into a chair just a few feet away from me in the skinniest blue jeans and a loose, half-sleeved blue silk top, thin gold hoops dangling from her ears, sipping from a huge coffee Thermos, nibbling at sliced avocado and rice crackers, I feel like I’m inhaling purified essence of Diaz. It’s making me dizzy.

She’s beautiful without being too pretty: her huge blue eyes are set into a face that’s almost too wide, above a smile that’s almost too cavernous, the whole engaging kit and caboodle offset by a nose that’s been battered and bent like an East End pugilist’s. Sure, she’s a babe, and she looks effortlessly fabulous in anything from a rubber wetsuit, when she’s surfing, to a strapless John Galliano Dior dress, when she’s presenting an Oscar. But Cameron Diaz is, above all, a great broad. She can beat you at pool, down tequila shots, burp, swear worse than Jonathan Ross, laugh uproariously at your bad jokes and, for good mea­sure, openly acknowledge that “Sex is the best!”. Which has endeared her to a never-ending posse of famous and equally good-looking boyfriends, including Matt Dillon, Justin Timberlake, whom she dated for four years, John Mayer, the surfer Kelly Slater and, most recently, the British model Paul Sculfor, from whom she has reportedly now split. Oh, and she’s a tireless fighter for the environment. What’s not to love?

A few years ago, not long after she’d starred in There’s Something About Mary — the quintessential Cameron Diaz role — The New York Times dissected her appeal with rather more historical, cinematographic precision. “Ms Diaz is a great American type, a game, outdoorsy girl in the tradition of Carole Lombard, Jean Arthur, Jane Russell and Angie Dickinson.” The writer noted that all these actresses had been discovered by the director Howard Hawks. “Ms Diaz would have fit marvellously into the tradition of Hawksian woman, with her sense of fun, camaraderie and forthright sexuality.” Not that any of Hawks’s actresses were required to sport the unorthodox gel that adorned Diaz’s hair in There’s Something About Mary.

Diaz has never disavowed the natural appeal — the “fun, camaraderie and forthright sexuality” — that has made her such a huge star. Why would she? For as long as age and her audience will allow, Diaz is going to keep doing films such as What Happens in Vegas, a typically rambunctious Diazian romantic comedy, which she made last year with Ashton Kutcher. Despite dire reviews, it took $220m at the box office worldwide, proof positive of her appeal.

Yet how long will that be? As she talks, I get the distinct impression, particularly now she’s moving into her late thirties, that Diaz has started to chafe at the strictures of feeling typecast as Hollywood’s sexiest, goofiest broad. Even if she denies this. “I don’t feel like anybody’s cheating me out of anything because of how they perceive me,” she tells me. Well, that I believe. She may have perfected the adorable goofball, but there’s a steeliness and determination to her that I hadn’t bargained on. “I live my life for myself and I am really happy with everything I have gotten to do.”

Diaz seems more in control of her own destiny than most Hollywood actresses, working only when she really wants to, using the rest of her time to do what she can for her environmental concerns — desperately trying not to be annoying and preachy about it — and trying to lead as normal a life as the paparazzi will allow. She has become tenacious in pursuit of her privacy and dogged in attacking and suing newspapers and magazines that lie about her. She admits that the attentions of the paparazzi became so oppressive for a while, particularly while she was dating Timberlake, that there were even times when she thought of abandoning Hollywood.

It helped somewhat when she bought a flat in New York, where she now spends as much as half her time.

Whatever she says, though, Diaz, using her leverage at the top of the Hollywood totem pole, is increasingly drawn to rather more serious roles that call on her to be much less who she naturally is than people she might have been had her life taken a different turn.

It’s not an easy transition, because we love the Cameron Diaz we have, and we will fight to keep her. A couple of years ago, she tried to venture beyond her comfort zone, and perhaps ours, in the family drama In Her Shoes, in which she played a troubled younger sister. Although the film was well reviewed, it was not a hit, taking just $35m in America. Some felt that audiences just didn’t want to see Diaz in a more serious role in a more serious film.

It will be fascinating to see how critics and audiences react to her new film, My Sister’s Keeper, which is adapted from the bestselling novel by Jodi Picoult. The role is an even bigger leap for Diaz. For the first time, she plays a mother; one of her three children, Kate, her teenage daughter, has leukaemia. Her younger daughter, Anna (played by Abigail Breslin), was specifically bred and, until she starts to object, is being used for bone marrow and blood transfusions to keep Kate alive. As Sara, the mother, Diaz faces the dilemma of balancing the very different needs of her two children: Kate, who will die unless Anna continues to be her biological lifesaver, and Anna, who wants control of her own body and life.

My Sister’s Keeper is directed by Nick Cassavetes, who directed The Notebook, and the film, a four-hankie weepie, is clearly targeted at much the same predominantly female audience as his earlier film. Playing Sara is the kind of Oscar-aimed role, requiring Diaz to wear little or no make-up, and even appear shaven-headed, that many actresses would fight for. More to the point, for an actress who is used to engaging the audience effortlessly, Diaz has to portray a character for whom we have less and less sympathy as the movie progresses. Sara’s instinct as a mother is to subject Anna’s needs to those of Kate, her dying sister.

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