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 how 3 cones fit 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 are made to perfectly fit each other 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?
We get the same ratio of 2 3 between the sphere's and cylinder's surface area.
But the same idea does not work for the cone.