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Surface Area of a Sphere

Date: 03/25/99 at 09:03:01
From: Gillian Bravo
Subject: Surface area of a sphere

How do I calculate the surface area of a sphere? There is a formula, 
but I can't remember it.

Date: 03/25/99 at 11:32:11
From: Doctor Bruce
Subject: Re: Surface area of a sphere

Hello Gillian,

The surface area of a sphere is  4*Pi*r^2,  where  r  stands for the 
radius of the sphere, and Pi is the constant approximately equal to 
3.14159. The symbol `r^2' means r x r; i.e., you multiply the radius 
by itself. 

You might recognize the quantity  Pi*r^2  (without the '4') as being 
the area of a circle of radius  r.  If you slice the sphere through 
the center you expose a circular surface which has area  Pi*r^2. So, 
the formula I gave you says that the surface area of a sphere is 4 
times the area of a circular cross-section through the center of the 

I once saw an experiment with a sphere of styrofoam. Someone sliced it 
through the center, then took one of the hemispheres and put a pin in 
the center of the cross-section. Then he wound string around the pin, 
laying it down smoothly the way you would coil up a garden hose, until 
the circle was filled up. Then he measured how long a piece of string 
it took to do that.

Then he did the same thing with the *outside* of the sphere, putting 
a pin exactly in the "north pole" and winding string around it until 
it covered the surface of the hemisphere. It turned out to take 
exactly twice as much string to cover the outer surface as it did to 
cover the flat cross-section. It seems that a hemisphere has twice the 
area of the cross-section. Since a whole sphere has two hemispheres, 
its surface area would be 4 times the area of the cross section.

Of course, the formula 4*Pi*r^2 is derived mathematically, using 
principles of calculus. The experiment with string is what we call 
'heuristic', meaning it is helpful to our understanding, but it is not 
a rigorous computation.

For more about sphere formulas, you can look at the Dr. Math FAQ:   

Happy surfacing,

- Doctor Bruce, The Math Forum   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry

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