Thinking about volume...

Question

zepdrix · Answer

Looking at the equation for volume of a sphere: $\large\rm v_{\circ}=\dfrac{4}{3}\pi r^3$

This value 4/3 seemed kind of ... random to me.
But I remembered that the volume of a cone is given by: $\large\rm v=\dfrac{1}{3}\pi r^2h$

So if we look at the cone with height equal to radius we have: $\large\rm v=\dfrac{1}{3}\pi r^3$
Which is precisely a fourth of the volume of a sphere!

zepdrix · Answer

So hmm,
I'm wondering if there is some clever way to deform a cone,
(Like cutting it in a clever way or something)
to show that it's shape is the same as half of a hemisphere.

zepdrix · Answer

@dan815 @Empty just a weird thought :)

empty · Answer

I have heard that this relation was first discovered by Archimedes and was put on his tombstone! I am interested in trying to figure it out, but I need to think a little.

empty · Answer

I'm thinking about like putting 4 cones in this awkward shape or playing with them |dw:1445420363944:dw|

empty · Answer

Cause that's the same volume as a sphere rearranged I guess, sorta odd.

zepdrix · Answer

That idea seems clever and intuitive,
I know that's how I would think of a square pyramid in relation to a cube.

empty · Answer

Yeah I don't really see this getting us anywhere I just needed to stick the 4 cones up there in as symmetrical of a way as possible but if I had to I'd use 6 cones in a cube. So weird. hmm

zepdrix · Answer

The great circle around the sphere of radius r has circumference 2pi*r.
Which is the same as the base of the cone.

What kind of shape do we get if we cut a cone along it's diagonal?
I can't seem to visualize it... if we unfold it, is it half of a circle?

empty · Answer

Oh wait, what if we place the tips of the cone at the center, and have them so that they are at the points of a regular tetrahedron?

empty · Answer

|dw:1445420730474:dw| I am trying to get as much symmetry as possible since a sphere as a lot of it, it will help to simplify the idea of deforming this in a single kind of a step but... this is admittedly pretty weird to imagine, it's just sorta some idea, but I think we gotta like chop it up somehow so idk