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I suppose they could split into 2 groups and use the shadow method from different latitudes.

Maybe they could also use the stars on the horizon at two different latitudes.

Is measuring a ship below the horizon feasible ?

What other methods could be tried by high school students?

Kantura
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    analyzing photographs of lunar eclipses doesn't require travel, though it requires math and assumptions https://youtu.be/hLPPE3_DVCw – uhoh Jan 18 '20 at 02:19
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    Great , thank you – Kantura Jan 18 '20 at 12:10
  • I vaguely remember having heard of the following method: sit on the beach, watch the sun hit the horizon, jump up and measure the time for the sun to hit the horizon again. Measure the height of your eyes sitting and standing. I tried but couldn’t immediately come up with a proper calculation. I don’t even know if it actually works so I didn’t make it an answer. Maybe somebody else can work out the details an earn the credits. – Hartmut Braun Jan 22 '20 at 10:13

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It is hard. You should consider what you are willing to allow them to research and what you will ask them to calculate.

First consider the classic "Shadow of a stick" method: On a certain day you measure the length of a shadow cast by a stick at two locations. The locations should be at the same longitude. By some trigonometry you can work out the number of degrees of latitude between the locations. By combining this with the distance between the locations you can calculate the radius of the Earth.

The biggest problem is how will your students know the distance between your chosen locations? Traditionally this would be done by surveying. The process of surveying is a lot harder than measuring the length of a shadow. If your students are allowed to google for the distance, they might (reasonably) argue that they could have just googled for the Earth's radius.

Moreover if students are actually going to use this method, they need to travel thousands of km North or South, this may not be practical, so your students can't actually measure the length of the two shadows.

A second method uses time. The time of sunrise (or sunset, or noon) is measured at two locations on the same latitude. This lets you find the angular distance between the locations and that can be used with the actual distance to find the radius of the Earth. Again this requires you to know the distance between the two points. And there are the same difficulties in travelling.

Looking at the shadow of the Earth as it crosses the moon can be used to calculate the radius. The is described in a document "Measuring the size of the shadow of the Earth" The difficulty with this method is that it depends on knowledge of the size of the moon (and the distance of the sun, though this can be approximated as infinity) This knowledge is difficult to get directly. However, this can demonstrate that the radius is finite and fixed, ie "the Earth is round".

In other words, to calculate the radius of the Earth, one needs additional information that can't be easily measured: Either the distance of two distant locations, or knowledge of the size of other astronomical bodies.

James K
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    Perhaps instead of traveling, students at two widely separate schools could collaborate. – Mike G Jan 18 '20 at 13:37
  • Sure they can collaborate with anyone. The problem is that they are not working independently, but are dependent on someone else's results. Is this any different from using google to find the length of the shadow of a stick. And is that any different from using google to find the radius of the Earth? Perhaps it is different. Which is why I start my answer with the caveat that "it depends what you will allow them to research..." – James K Jan 18 '20 at 15:11
  • "same longitude" isn't necessary if you just make several measurements between 10:30 and 13:30 and find the minimum, and size of the moon could be replaced with distance since the moon is acting as a sort of screen. – uhoh Jan 19 '20 at 01:24
  • Agree with both of those. But you still need the distance (not easily measured). And the Lunar distance is just as difficult as the Lunar radius (indeed, if you know one it is easy to find the other) – James K Jan 19 '20 at 18:54
  • Right - which is why that, while they are in error, I can at least sympathize (see my username) with flat Earth believers: actually and truly independently demonstrating even this "simple" 2300-year old result is quite non-trivial, and ultimately, unless you're going to do all that work and spend all that money to travel, like with many other things, you have to appeal to authority. Hence, I can understand how those mistrustful of such, can be led to believe this way. And I think that people need to be more honest about what they do and do not believe as coming from authority, and how that – The_Sympathizer Jan 26 '20 at 06:29
  • there really aren't any truly "free" thinkers. – The_Sympathizer Jan 26 '20 at 06:30