Measuring the Radius of the Sun

How does one measure the radius of the Earth using...the Sun?

The answer's quite simple, in theory. You'll need at least two people, a stopwatch,  and a clear view of the horizon.

Last Sunday, the class went on a trip to Santa Monica beach to put theory to practice. We split up into groups and attempted to calculate the radius of the Earth using only the sunset and stopwatches. My group consisted of myself, Ronnel, Andy, and Krishnan.

Here's how we did it. Consider the Earth, modeled as a sphere. Say you have two people, one located at the surface, and one located a height 'h' above the surface. Their views of the horizon (the line of sight tangent to the surface of the Earth) will thus be different:

By geometric analysis, you can easily find that the angle made between your different horizons corresponds to the angle made between radii drawn fromthe Earth to the points on the horizons where the line of sight is tangent to the Earth (see above diagram). Using simple trigonometry, we get:

How do we find theta? It's just the angle between the two horizons - therefore, if you measure the difference in time between sunset for both observers and multiply by the angular speed of the Sun across the sky, you've found it. So:

Combining both equations, you get:

So now we have an expression for R. Let's plug in some numbers. The Sun travels 180 degrees (across the sky) in twelve hours, so it's angular speed is:

When we measured the Sun, we took multiple measurements using 4 people located at different heights. Our best time difference was 25.2 seconds, for a height difference of 3.33 meters. So:

Plugging this all in, we have:

The actual radius of the Earth is about 6378km, so we're off about 69% percent - yikes! But at least we're correct to within an order of magnitude, which is all you can really expect with a measurement this crude.

Science!