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#1 2017-10-02 22:16:48

Viki1177
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geometry

Why is the cone one third of the cylinder if a cone and cylinder have the same radius and height ?

#2 2017-10-02 22:40:15

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,306

Re: geometry

Hi,

The volume of a cone is

cubic units and the volume of a cylinder is
cubic units where
, r is the radius, and h is the height in units.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2017-10-02 23:57:52

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,139

Re: geometry

hi Viki1177

Welcome to the forum.

If a cone and a cylinder have the same radius and height, you'd expect the volume of the cone to be less than the cylinder.  But why one third and not some other fraction, or maybe even an amount unconnected with the cylinder's volume?

I can show this if you know some calculus; specifically a little bit of integration. Post back if you want to see this.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2017-10-03 01:38:44

iamaditya
Member
From: Planet Mars
Registered: 2016-11-15
Posts: 725

Re: geometry

Oh wow,

It seems ages since Ganesh has posted something in Help Me. He usually used to post in Ganeh's Puzzles and Dark Discussions at Cafe Infinity.

If a cone and a cylinder have the same radius and height, you'd expect the volume of the cone to be less than the cylinder.  But why one third and not some other fraction, or maybe even an amount unconnected with the cylinder's volume?
I can show this if you know some calculus; specifically a little bit of integration. Post back if you want to see this.

Yeah, I too want to know about it.

Last edited by iamaditya (2017-10-03 01:39:52)


Practice makes a man perfect.
There is no substitute to hard work
All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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#5 2017-10-03 20:33:57

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,139

Re: geometry

hi iamaditya

This result is true for any prism and cone which have the same cross-section / base.  Curiously it's easier to prove in the general case than for a specific cross-section so I'll do that.

Sr4RzbO.gif

The first picture shows a prism which has a completely irregular, wiggly perimeter.  The second shows the same cross-section; this time forming the base of a pyramid.

Firstly: a general principle for calculating a volume. If you know the area of a shape and then extend it in a third dimension at right angles to the base then the volume of this solid will be area of shape times the height of the extension.  Imagine the area divided into square centimetres.  Each square makes a cuboid when extended so the result is true for these.  If the shape is irregular, make as many square centimetres as you can, and then fit square millimetres for the rest.  The result will be true for these square millimetres.  Continue to sub-divide any remaining area into smaller and smaller squares and the result continues to be true.  Hence it is true whatever the start area.

Now for the prism calculation:

Imagine the prism is divided into many small slices, each having the same cross-sectional area, A and 'height' delta x, a little bit in the x direction


Now for the pyramid.

The defining property for a pyramid is that the cross-section stays the same shape as the base but with a steadily diminishing size.  If the base has area A, then a cross-section at height x (measured from the vertex) will have an area that is proportional to the area of the base.  Say the base is at x = h and the vertex is at x = 0.  The cross-section at height x is a shape that is similar to the base.  The lengths are in the ratio h:x, so the areas will be in the ratio h^2 : x^2.

Hence


So the volume of the pyramid is one third the volume of the prism.  The result in post 1 is just a special case where the cross-section is a circle.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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