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## Failure Mode:  Spherical Geometry, Geometry

Here is the YouTube video in question:
Flat Earth BUSTS Google Earth! (Prove It Yourself)

One thing I want to do is remind myself and readers that perhaps this guy is really trying his best to understand things - to that end I hope my comments serve as constructive criticism.

But so many things go wrong in this short video it is difficult to know where to start.

First of all, why pick 8.5" for the base of his triangle?  That just seems arbitrary -- which is fine -- any length will work but it sure makes everything else quite confusing and difficult to work with because the sides are similar in length (but not exactly) and the angles can easily be made to look confusing.  Which I hope is what throws him off.  But if you want to make all the angles 60° you would need to make the base very short indeed, as we find out next...

Secondly, "the top angle is much larger than 60°" -- that is absolutely correct and that is because it's on a sphere where the angles simply must add up to greater than 180°.  Indeed, it is mathematically impossible for those three angles to be 60°.  The closer you make all three angles to 60° the smaller the area would be -- at exactly 60° the area would equal 0 - ZERO - Zilch - Zip - Nada - which would then make it a point and not a triangle. ONLY in ideal Euclidean (flat) space do the three angles of a triangle equal exactly 180 degrees.

Here is what it actually looks like when you draw his big triangle inside Google Earth (my end point are only approximate but in roughly the same area).  Each line is the shortest distance on the surface from point-to-point.  They "bow" out here because of perspective - which again, happens because it is a sphere.

To prove unequivocally that Google Earth is using Spherical Coordinates here let's draw a bigger triangle that is easier to work with.  We're going to go from the North Pole to 0° N/0° E to 0° N/90° E and back to the North Pole.  That gives us nice round numbers to work with and we can clearly show Google Earth is using an Oblate Spheroid model.  You can see it still looks basically the same, just a little bigger.

Google Earth gives us two key values here:

Perimeter: 30,023 km
Area: 63,758,207 km²

Now if you think about that figure for a minute you'll realize that we have two arcs from the North Pole to the Equator, each of those is therefore 1/4 of the Polar circumference of the Earth - and one Arc from 0° to 90° along the Equator which is 1/4 of the Equatorial circumference of the Earth.  This obviously makes it so much easier to do the math.

We can even feed that equation into Wolfram|Alpha and let it do the calculation (I've provided several handy direct links so you don't have to trust my math).

[(earth polar circumference in km)/2] + [(earth equatorial circumference in km)/4] = 29,989.081 km

We also find the length of each arc is:
earth equatorial circumference in km/4 = 10018.754 km
earth polar circumference in km/4 = 9985.1631855 km

That's pretty close (0.113% difference) but should it be exact?   Well, no, actually!

Google Earth uses much more exact method of calculating the distances using the WGS 84 Geodetic model (which is the reference grid for GPS as well) which gives results accurate to about half a millimeter.  But our quick swag wasn't too bad and it's certainly close enough to confirm these are not merely straight-line distances.  We amateur cartographers need to understand that when we work with values like radius and circumference these are averaged over the whole Earth.  If you added in all the corrections from WGS 84 that Google Earth does you would get the same value.  The Google Earth values are much more accurate than I need to be.

I'm just trying to break it down to a close, but not exact, model so we can understand the math here - that way we can see which one is the better fit.

Problem for the Reader: What is the straight-line distance from the Pole to a point on the Equator?
Hint: It makes a right angle with the center of the Earth and you can use the Pythagorean theorem to solve this using the polar and equatorial radius of the Earth to solve for the hypotenuse.

So we can already see the claim that Google Earth is using a "flat map" is plainly false, but it gets so much worse when you try to compute the area.

Since we know the approximate lengths of each side, let's see what we get for area of a Euclidean triangleFor an arbitrary triangle the area is given by:

You can verify that with a simple case:

[(a+b-c)(a-b+c)(-a+b+c)(a+b+c)]/4 where a=5, b=4, c=3 => 6 -- checks out

Taking these line segments as the lengths of our triangle:

earth equatorial circumference in km/4 = 10018.754 km
earth polar circumference in km/4 = 9985.1631855 km

[(a+b-c)(a-b+c)(-a+b+c)(a+b+c)]/4 where a=9985.1631855, b=9985.1631855, c=10018.754

Gives us 43,269,400 km² NOT EVEN CLOSE.  Treating these values as a flat area simply cannot work, you aren't even in the ballpark.  You have to treat it as a spherical section.  The formula for the area of a spherical triangle is pretty simple - we just need the angles and the radius to get an estimate (again, Google uses a much more accurate model of the exact oblate spheroid - we're going to see if a simple sphere is close enough -- this works because the oblateness of the Earth is fairly slight).

For a sphere the equation is:

ε = α + β + γ - π
A = ε r²

Which gives us 63,686,900 km² -- which again, is close to Google Earth value where flat model is FAR off.

And finally, here is what the corner looks like for each vertex of our spherical triangle - this is approximately a 90° angle -- since we have 3 vertices that adds up to 270°, clearly spherical geometry here as well.

So by every angle the flat model fails - the lengths are spherical, the area is spherical, and the angles are spherical.

The smaller you make that area the closer it will appear to be flat and the harder it will be to measure and confirm what you are seeing.  But this is only an appearance and the deviation from flat exactly matches the spherical Earth model Every Single Time.