Cone vs Sphere vs Cylinder
Volume of a Cone vs Cylinder
The volume formulas for cones and cylinders are very similar:
|The volume of a cylinder is:||π × r2 × h|
|The volume of a cone is:||1 3 π × r2 × h|
So the cone's volume is exactly one third ( 1 3 ) of a cylinder's volume.
(Try to imagine 3 cones fitting inside a cylinder, if you can!)
Volume of a Sphere vs Cylinder
Now let's fit a cylinder around a sphere .
We must now make the cylinder's height 2r so the sphere fits perfectly inside.
|The volume of the cylinder is:||π × r2 × h = 2 π × r3|
|The volume of the sphere is:||4 3 π × r3|
So the sphere's volume is 4 3 vs 2 for the cylinder
Or more simply the sphere's volume is 2 3 of the cylinder's volume!
And so we get this amazing thing that the volume of a cone and sphere together make a cylinder (assuming they fit each other perfectly, so h=2r):
Isn't mathematics wonderful?
Question: what is the relationship between the volume of a cone and half a sphere (a hemisphere)?
What about their surface areas?
No, it does not work for the cone.
But we do get the same relationship for the sphere and cylinder (2 3 vs 1)
And there is another interesting thing: if we remove the two ends of the cylinder then its surface area is exactly the same as the sphere:
Which means that we could reshape a cylinder (of height 2r and without its ends) to fit perfectly on a sphere (of radius r):
(Research "Archimedes' Hat-Box Theorem" to learn more.)