What is the relationship between cylinder volume and circumference?

Question

yesterday, I was asked how could I guess the circumference of a beaker (cylinder) given that the volume was 150 mL(cm3). I'm having trouble finding a relationship. I tried to incorporate calculus, using C=2pi(r) and V=2hPi(r)^2, but i could not put them in terms of each other. Any other suggestions? thank you!

 

Answer 1

You can express the circumference in terms of volume only if you know the height of the cylinder. Two cylinders may have same volume and different circumference of the base, because they may have different heights. Think about two cylindrical jars with the same volume, but one of them with larger base. It will have shorter height.

 

Your volume formula is not correct.

 

V = Pi•r²•h

 

You CAN solve this formula for radius, and then multiply the radius by 2Pi to find the circumference.

 

r =√ [V/(Pi•h)]

 

The expression in the brackets is the radicand.

 

C = 2•Pi•√ [V/(Pi•h)].

Answer 2

C = 2·pi·r

C² = 2²·pi²·r² = 4pi²r²

 

V = pi·r²·h = C²·h/4pi

 

Or, solving for C:

 

√(4pi·V/h) = C