# Measure the radius of the earth using the sun?

Can someone tell me why this would or would not work?

I have a sun spotter device that projects an image of the sun onto a piece of paper using a lens and some mirrors. When you use the device it’s quite clear that the sun moves across the image pretty quickly due to the earths rotation. This got me thinking: I can measure (on the paper) how many inches the sun shifts over some timescale, and divide that timescale by 24 hours. This should be the angle that the earth turned through. Then i can use s = r theta and solve for r, where s is the length that i measured and theta is the angle. Should this work?

I’m unsure because the earth rotates quite quickly, and the image of the sun only shifts slightly. Perhaps the lens makes things difficult

• It's not clear what your sun spotter is doing. A diagram would help. Is it like this? scientificsonline.com/product/sunspotter Apr 12 at 4:50
• "the earth rotates quite quickly" Indeed! At the equator, the rotation speed is ~465 m/s, and about half that at latitude 60°. Obviously, your sun spotter image, or shadows on a sundial, don't move at that speed. ;) Apr 12 at 4:54
• This seems like a very simple experiment to run. When you did it, what did you come up with? Did your answer make sense? Apr 12 at 16:12
• You've essentially made a sundial. You could, in theory, measure the angle the Sun moved and measure the distance the image moves, but doing so over a short distance is going to have a lot of error margin. It is essentially what Eratosthenes did, but over a much, much larger distance. Apr 12 at 16:29

## 1 Answer

No, because the distance that the sun moves on the paper (s) is not the length of the arc that you have moved as the Earth rotates. It is the length of an arc with centre at the sun spotter and radius = distance from sun spotter to paper.

Your formula would only find how far your sun spotter is from the paper on which you are projecting the sun. For example if s=3 inches and theta= 15 degrees (or 0.26 rad) then you would find r= 11.5 inches, and that is obviously not the radius of the Earth!

• Although you haven't invented a way to measure the Earth, you have invented an excellent clock, unfortunately the ancient Egyptians and Babylonians got there first. Apr 12 at 5:24
• You can determine the radius of the Earth from observations of the Sun. Look up Eratosthenes online. I just looked and got over 5 million results. He got a number very close to what we now know. His math was sound but he was lucky in having his measurement errors cancel one another. Apr 12 at 13:10
• While yes, that's a very different method. If I have my ancient greeks straight, then he put two equal-length sticks in the ground vertically, and measured shadow lengths on the longest day of the year. Most relevantly, he did this at different latitudes, then calculated with triangles. It's fundamentally impossible to use that method when in one location. Apr 12 at 13:43
• So it is possible to do what the question title says, but not with the method in the question. @stretch Apr 12 at 15:31
• @Barmar It's not possible to determine the radius of the Earth by observations in only one location. You need to observe from two different locations to determine the radius. Apr 13 at 21:00