Throw in one part Madonna’s “Truth of Dare” and one part Miley Cyrus’ “Hannah Montana: Best of Both Worlds” and one part Zooey Deschanel’s “New Girl,” add a pinch of the original “Willy Wonka and the Chocolate Factory” and a smidge of Tim Burton’s unfortunate 3D fantasia “Alice in Wonderland,” stir, bake accompanied by an infectious pop soundtrack…and voila…you have Katy Perry’s new concert documentary “Part of Me.”

We all more or less already knew Katy was an intensely likable personality with a knack for marrying catchy melody and zany “coquette-in-Candyland” visuals, but we probably didn’t realize how deep-feeling she could be or how sad and challenged her life had been. Yes, the film, like “Truth or Dare” before it, is a calculated play to humanize (and expand the brand of) its central pop heroine. Unlike that film, there is an authenticity to Perry (benefiting no doubt from 20+ years of us all living out loud, online, and through the self-aware guise of reality TV) that Madonna couldn’t/can’t effect.

Your heart genuinely breaks for Perry when the de facto villain of the piece Russell Brand (standing in for Warren Beatty from the Madonna film) ends their marriage. (One of her handlers remarks at one point, “Katy keeps leaving the tour to go see him…when is he going to ever travel to her?”) You also wonder how overbearing her fundamentalist religious upbringing must have been when you meet her traveling minister mom and dad who resemble even scarier versions of Sharon Osbourne and Swifty Lazar. Finally you leave the theatre with an uplift when Katy “conquers” all to sing triumphant versions of her hits “Firework” and “California Gurls.” (Oh-kay, that last bit may be a bit overstated since she is dressed like a giant peppermint at the time.)

The film is a lot of fun, and, yes, a bit contrived…and completely unnecessary to view in up-charged 3D. (2D will do just fine, thank you very much.) All in all, though, it is a fair and fizzy assessment of a pop star on the ascent, one of the few for whom you genuinely wish a happy and successful life. Just be prepared (with earplugs) for the shrieks that may emanate from some of your fellow audience members (the 12 and under crowd) when Justin Bieber makes the requisite appearance onscreen. Ugh.