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**Finding the shape of the Earth using Polaris**

We shall here explore the position of the star Polaris, and what implications this has on the shape of the earth. We first have several facts to consider:

- The star Polaris is almost directly above the Earth’s north pole.
- The angle of elevation from the ground to the star Polaris = the latitude.
- Polaris can only be seen from North of the equator (or slightly to the south of it depending on the observer’s height).
- Light travels in a virtually straight line from a light source to the observer.
- The distance from the North Pole to the equator is about 10018 km or 6215 miles (If you Google this you might get slightly different values, but they are just different by 0.1 % or so).
- A change of 1 degree of latitude = a change of the above value divided by 90. I.e., the latitudes are equally spaced.

**Triangulation**

It ought to be easy to find the elevation of Polaris. Just measure the angle of elevation of Polaris, then there is just a question of simple geometry.

Say we are at latitude 45 degrees, then we are about 5000 km, or 3100 miles from the North Pole. That would indicate that Polaris would be at 5000 km, or 3100 miles above the horizon.

Let us try triangulation again, but now from a latitude of 60 degrees. Each degree is about 10018/90=111 km apart, or 6215/90 = 69 miles. At 60 degrees you are 10018-10018·60/90 or about 3300 km, or 2100 miles away from the North Pole. Then we have that h/r=tan of 60 degrees. That gives us a height of Polaris of about 5700 km or 3500 miles.

So, this would give us two different heights of Polaris. And every other latitude would give another altitude. This would be true for all other stars too. Triangulations would give different heights depending on the positions used.

The question for a flat earther would then be – what of the underlying assumptions must be wrong? I guess the most common “explanation” would be “perspective”, but then ask yourself, what paths must the light take from a particular position of Polaris for it to reach the Earth in the observed angles?

You can try in the figure below. Polaris is placed where light moving in a straight line would reach the ground at the correct angle at a latitude of 45 degrees. Now try with the other points. Could you make straight-line light beams hit the ground at the observed angles? If not then, why not?

**The simulation**

On the page linked below, you can explore where Polaris must be, and what radius the earth must have. For each 10-degree increment of latitude the position Polaris is indicated. That is, it shows how far above the Earth Polaris has to be to be directly above the North Pole.

The blue line is the surface of the Earth. There are green lines indicating every 10^{th} latitude line. The yellow lines correspond to the light coming from Polaris to the Earth. The angle of elevation between a yellow line and the ground is always equal to the angle of elevation of Polaris. The red circles are the necessary positions of Polaris to enable the above.

You can change the radius of the Earth using the Up and down arrows. You can also click, and hold down a mouse button, above or below the “earth” or touch the screen above or below the “earth”.

The current radius is shown in both miles and kilometres.

The only radius that does not require multiple positions of Polaris is the one corresponding to the official radius of the Earth. The light beams will now be virtually parallel, corresponding to a Polaris that is far, far away.

Observe that the radius of the Earth is not assumed. It will be a conclusion drawn from the facts stated in the beginning.

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