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#1 2006-03-26 21:11:44

MathsIsFun
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Showing that a Cone has one third the Volume of a Cylinder

I was recently asked this question by email "How would you prove that a Cone has one third the Volume of a Cylinder", and I responded like this:

I wrote:

Using Calculus:

Firstly, the cylinder is easy, just have lots of little discs, each disc has a radius of "r" and an area of  πr, and we need to integrate over height:

V = ∫πr dh = πrh

That was fairly easy!

Now for the cone, the little disc's radius get smaller as you get higher. The rate they get smaller is a constant, the slope of the sides, which we can call s.

For simplicity I will turn the cone upside-down, so the disc's radius get bigger with height. Each disc will have a radius of "sh" and an area of  π(sh), and we need to integrate over height:

V = ∫π(sh) dh = ∫πsh dh = πs (1/3)h

Now we know that the slope s = r/h, so: V = π(r/h) (1/3)h = πr (1/3)h

Which is one third of the cylinder's volume!

Now, I recall seeing proofs without calculus that did something similar, they turn the cone into "N" small discs, and add them up.

Disc number "i" will have a height of h/N, and a radius of r(i/N), with a volume of πr(i/N)(h/N) = πrih/N

Summing over "i" : V = ∑πrih/N = πrh/N ∑i

The only thing in the way now is "∑i", which I know can be simplified to a few terms, but am not sure how. If that could be done, hopefully we would see the formula come out to be πr (1/3)h

So, I haven't totally solved this for you, but I hope I have helped.

I would like to do a page about this, can someone correct my work and give an easy way to complete the ∑i term?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman
 

#2 2006-04-06 23:53:41

krassi_holmz
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Re: Showing that a Cone has one third the Volume of a Cylinder

good.
The sum usually is proven by induction, but here's a direct way :
http://www.mathsisfun.com/forum/viewtopic.php?id=3146


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