Surface Area of Earth (a Sphere)

Date: 8/30/96 at 13:18:48
From: Melanie
Subject: Surface Area of Earth (a Sphere)

Dear Dr. Math,

Could you tell me the formula for determining the surface area of a 
sphere?  I need to find the surface area of the earth based on its 
radius and I can't seem to find this formula in my physics text.

Thank you,
From a very grateful engineering student,
Melanie

Date: 8/30/96 at 15:37:15
From: Doctor Alain
Subject: Re: Surface Area of Earth (a Sphere)

The surface area of a sphere is given by A = 4 * Pi * R^2.

This will give a good answer for most purposes, but Earth is not a 
sphere. Earth is a little flat at the poles (the distance from the 
center to a pole is shorter than the distance from center to anywhere 
on the equator) and is a little rough. 

The fact that it is flat at the poles doesn't change much its surface 
area, but its roughness does. If you take a flat square field 100 
meters on its side, its area is about 10 000 m^2 (not exactly because 
of the curvature of the surface, but very close) now if there is a 100 
meters high pyramid shaped hill on a square lot, 100 meters on its 
side, the surface area of the hill is much more than 10 000 m^2, it is 
about 22360 m^2, so if all the surface of the Earth was covered with 
such hills the surface area of Earth would be more than twice what is 
given by 4 * Pi * R^2. But in fact, even the 100 m high hill should 
have a surface area much greater than 22360 m^2 because hills are 
themselves rough. So depending on the scale at which you choose view 
the surface, the area will change. The smaller the scale the bigger 
the total area.

You can always increase the total area by considering ever smaller 
pits and humps, all the way until the surface has no more meaning (if 
you look at the surface at a scale smaller than the distance between 
atoms, Earth is no longer a single object, and measuring the surface 
of subatomic particles can lead to meaningless things). But if you 
look at Earth from space, you will see that it is quite smooth, so you 
can compute its surface area with A = 4 * Pi * R^2 and forget about 
the extra area given by mountains, hills and humps.

-Doctor Alain,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 8/30/96 at 16:55:17
From: Doctor Jerry
Subject: Re: Surface Area of Earth (a Sphere)

The surface area of a sphere of radius a is 4*pi*a^2.

One way of remembering this: the area of a circle of radius a is 
pi a^2.  If you differentiate this with respect to a, you get 2*pi*a, 
which is, gasp,  the circumference of a circle of radius a.  The 
volume of a sphere of radius a is 4*pi*a^3/3.  If you differentiate 
this with respect to a you get, yes, the surface area of a sphere. 

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/