The average time between transits of the Sun is exactly 24 hours (that is how the period of 24 hours was originally defined). The average time between transits of the Moon is 24 h 48 m 46 s.
This time interval was (and still is) very important to people who calculated tides. For a long time, it was called a "lunar day".
During a synodic month, this interval would add up, until the Moon had made a complete circle and had gone back to being in line with the Sun.
48 m 46 s = 0.812777...
The Moon "loses" 0.812777 hours per day. How many days before the Moon loses exactly 24 hours?
24 h / 0.812777 h/d = 29.5284 days
= 29 d 12 h 40 m 51 s
(The real value is 29 d 12 h 44 m 03 s)
To find the very good approximation, you only need an accuracy of one second for each observation and you only need to observe over a few months. To get the very accurate (exact) figure, you would need a better clock (with fractions of seconds) and observations covering a few years.
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A US dime (the coin) has a diameter of 0.705 in. (= 17.91 mm).
Hold the dime so that it exactly covers the full circle of the Moon (even if you don't see the entire disc of the Moon, you have to imagine the whole thing).
Have someone carefully measure the distance between you eye and the dime, when you exactly cover the Moon with the dime. You should get (roughly) a distance of 78 inches (six and a half feet, or roughly 2 metres.
Repeat this observation over many days, to make sure you get a good average (the distance to the Moon changes during the month, and you need the average).
This would give you your own measurement of the apparent diameter of the Moon.