Cone vs Sphere vs Cylinder

cone sphere cylinder

Volume of a Cone vs Cylinder

Let's fit a cylinder around a cone.

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!)

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.

cylinder vs sphere

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!

The Result

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):

Cone Sphere Cylinder Volumes

Isn't mathematics wonderful?

Question: what is the relationship between the volume of a cone and half a sphere (a hemisphere)?

 

Surface Area

What about their surface areas?

cone sphere cylinder area

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.