Her death was announced by the estate of the musical's creators, Rodgers & Hammerstein, on Monday.
She was diagnosed with brain cancer four weeks ago and died on Christmas Eve, news site TMZ quoted her son Ryan as saying.
"She was an actress, a ballerina and loved living her life to the fullest," he told TMZ.
Born Heather Menzies in Toronto, she was 15 when the musical film was released in 1965. It went on to win 10 Oscars, including best picture.