To work the weighing method also needs a standard of area, cut from the same sheet. Galileo was allegedly balancing the cycloid cutout with that of its generating circle, and found that they were as $3:1$, but did not believe it because he expected that they are incommensurable. The first association that comes to mind is Archimedes's Method, where he cuts cone, sphere and cylinder into thin slices, and then balances them on a lever. Of course, this was a purely mental exercise, but in the famous Eureka incident (if Vitruvius's tell tale is believed) Archimedes was presumably balancing the suspicious crown and a slab of gold on physical scales. Of course, that was to compare densities rather than areas or volumes, but the re-purposing is straightforward.
The Galileo story is not exactly an anecdote. It is reported by his loyal assistant (some say "desciple") Torricelli in Opera Geometrica (1644), only two years after Galileo's death, citing a 1639 letter of Galileo's to him. Torricelli and his student Viviani spent the last years of Galileo's life (1639-1642) helping him, and were two of the three people present at his deathbed, the third one was his son. However, after Galileo's death Viviani wrote his first biography, where he told another story, you heard of it, about the balls dropped from the leaning tower of Pisa. "He clearly embellished things a little...". It doesn't mean that the student took after the teacher, and Torricelli's cycloid story is also made up, but it does raise some doubts. Roberval, one of Torricelli's correspondents, found the area of cycloid geometrically back in 1634, so he likely knew what the correct ratio was, even without the Galileo's letter.
At a risk of giving material to conspiracy theorists I'll point out another quirk of history. Another "desciple" of Galileo's, Cavalieri, reinvented Archimedes's method of "indivisibles" in 1627, and in a streamlined form that did not require levers and balancing. In 1644, the same year that Torricelli published his cycloid story, Cavalieri used indivisibles to show that cycloid is to the generating circle as $3:1$. Torricelli lavished Cavalieri with praise for indivisibles, but wrote:
"I should not dare affirm that this geometry of indivisibles is actually a new discovery. I should rather believe that the ancient geometricians availed themselves of this method in order to discover the more difficult theorems, although in their demonstration they may have preferred another way, either to conceal the secret of their art or to afford no occasion for criticism by invidious detractors".
In this he was a prophet. In 1906 Heiberg found a palimpsest, later lost again and rediscovered with much hype in 1998, containing the only extant copy of Archimedes's Method. It is a letter to Eratosthenes, where he "availed himself of this method" of balancing those indivisible slices of cone, sphere and cylinder on a lever to discover their volume ratios. The method he "concealed" in On the Sphere and Cylinder by giving an unilluminating but impeccable exhaustion proof, "to afford no occasion for criticism". So far as we know neither Cavalieri, nor Torricelli, nor anybody else in the West knew of the Method before 1906. Sometimes history winks at us. ;-)