



 
Hi Bruce, I got the same expression for the area in the answer to an earlier question about a contact lens. (I used R where you used r.) If you look at what I did you will see that h units up from the bottom of the test tube the radius of the circular cap is √(2Rh  h^{2}) as long as h ≤ R. If h > R then the radius of the test tube is R. Since the circumference of a circle is 2 π r where r is the radius the circumference of the test tube at any point h units upward from the deepest point is
Harley
 


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