Officers from Dorset Police say they are "confident" the body is that of the teenager.
The discovery on land near Swanage was made at about 3pm on Saturday by specialist search teams close to the coastal path and the field where her clothes were found on Thursday.
Her sister Clara Pope said: "She is the absolute light of my life. So beautiful, so emotionally wise and intelligent and so passionate and artistic and creative... I will always, always, always be one of three."
The teenager's cousin, Marienna Pope-Weidemann, thanked those who helped in the search for Miss Pope, calling their efforts a "ray of light in this nightmare".
She added: "We are absolutely devastated and unable to put these feelings of loss into words. Our little bird has flown but will always be with us."
In a statement, Detective Superintendent Paul Kessell said the body was not yet formally identified.
He said: "Although the body has yet to be formally identified, we are confident that we have found Gaia.
"Her family have been informed and are being supported by specially trained officers. Our thoughts remain with all of her family and friends at this very traumatic time."