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Michael Lee's Mathematics and Physics

The Surface Area of a Circle Derivation

 

I'm going to show three methods here.  The first and second ones, use trigonometric substitution and is based on Cartesian coordinates involving lengthy, complicated calculations and the other involves simple geometry.

 

First Method Using Trigonometric Substitution

 

                     Take any circle of radius r;  hold r constant and note the following...

 

 

 

 

 

 

Now this is the tricky part...

 

 

 

 

                Here is the second method.