I learned as a child (and a common tale), that Aristarchus gave a method of measuring the sun-earth-moon angle to estimate the ratio of the earth-moon distance to the earth-sun distance. Indeed, during the first quarter moon phase, the sun-earth-moon reasonably form a right triangle at the moon.
I want to illustrate this to a math class that I am teaching, however I want some practical ways of measuring this sun-earth-moon angle. I reckon it will probably be a bad estimate anyway, as this angle might be nearly 90 degrees as the sun is quite far from the earth. But it might be a fun activity anyway. Are there any reasonable way to measure this angle with modest tools like a level and a plumb? (Or smart phones with a leveling tool.) Do I need to wait when both the sun and moon are visible in the sky during first quarter moon phase? Or, how did Aristarchus measured it, if he did at all?
Sorry I am not a practicing astronomer. Thanks for your insights.