A mathematical look at the shape of the Earth by David E.

Using math to demonstrate Earth's shape

Note: This analysis is by David Emery from the SciManDan Patreon Slack group, a big thanks to him for this content.

Earth’s surface area

Our interest here is in the surface area of the earth.  The question is how would the surface area of a spherical earth compare to a flat, circular disk earth.  It can be hypothesized that the sphere and the accepted flat earth would have the same or very close to the same surface areas because the flat earth is an Azimuthal Equidistant Projection of the sphere. 

Our two hypotheses here would be:

H0:         Surface area of a Sphere with Radius = 3958.8 miles (6371 km) is equal to the surface area of a flat circular disk of surface radius comparable to the sphere.

Ha:         The two surface areas are statistically different.

The radius of the earth is an approximation due to the earth being an oblate spheroid.  The actual radius varies from 3,950.06 miles to 3,963.11 miles (6,357 km to 6,378 km).  That gives us an ellipticity of 0.0033 or 0.3%.  That minute difference is just not enough to be concerned with in this analysis.

The radius of the circular flat earth is not a published number.  It is assumed that there is no desire to have undue focus on the overall size of the flat earth by any measure.  This radius is easily calculable using latitudes.  The equator is universally accepted as 0°.  The North Pole is defined to be 90° N and the South Pole (or southern edge) is defined to be 90° S.  The North Pole is the center of the flat earth disk.  Therefore, the radius of the flat earth disk is 180°.  Each degree of latitude is 69.38 miles (111.66 km).  The radius of the flat earth must be 12,488.4 miles (20,098.8 km).

Experiment

We need to look at the corresponding surface areas for the sphere earth and the flat earth.  This is the calculations.  The equations can be found in any elementary geometry book.

The surface area of the disk comparable to the flat earth is 2.5 times the surface area of the sphere comparable to the spherical earth.  

A mathematical simulation is good proof. The next step would be to find real life data to compare to the simulation.  There are many internet websites that have posted surface area data.  Individual countries across the world report total surface area each year. Data back to 1961 has been reported.  The following is a partial list of website URLs referenced: T

These totals compare favorably to the spherical surface totals shown above except that we have not yet considered the surface areas of oceans and seas.  The National Oceanic and Atmospheric Association (NOAA) keeps constant track of this and other scientific data about the ocean and atmosphere.  The latest data included surface areas of 18 oceans and major seas.  The summary is as follows:

We now have total surface areas for both land and water measured and reported by multiple sources and repeated over multiple years. Reality totals should match the totals from the math simulations completed earlier.  Totals are shown in the table below.

 Total Land Area (sq miles)Total Land Area (sq km)Total Water Area (sq miles)Total Water Area (sq km)Overall Total (sq miles)Overall Total (sq km)
Calculated Sphere    196,941,2185.10xe+8
Calculated Flat    489,962,79912.69xe+8
Measured Sphere Surface Area56,887,686147,338,490138,371,500358,381,200195,259,1865.06xe+8

It is clear, that the sphere math simulation is nearly the same the measured value.  The flat earth math is once again approximately 2.5 times bigger than the measured total.  There is no doubt that, in the total earth surface area is exactly that of a sphere. The flat earth is not even in question.

There is one final calculation that was a last-minute realization.  Water comprises 71% of the surface area of the earth.  The flat earth has no published measured values.  Take the surface area of water divided by the total earth surface area and you get 138,371,500/195,259,186 = 70.87%.  This is one more proof that the surface area of the earth can only be correct if earth is a sphere.

David Emery

B.S. Mathematics, M.S. Industrial Engineering 16 years as a NASA Statistical Analyst, University Statistical Analyst

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1 Comment

  1. Very cool mathematical proof, I love it! Not that it will stop the flat-earthers from trying to come up with a matching map anyway…

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