# Cone vs Sphere vs Cylinder

## Volume of a Cone vs Cylinder

Let's fit a cylinder around a cone.

The volume formulas for cones and cylinders are very similar:

The volume of a cylinder is: | π × r^{2 }× h |

The volume of a cone is: | \frac{1}{3} π × r^{2 }× h |

So the cone's volume is exactly one third ( \frac{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.

The volume of the cylinder is: | π × r^{2 }× h = 2 π r^{3} |

The volume of the sphere is: | \frac{4}{3} π × r^{3 } |

So the sphere's volume is \frac{4}{3} vs 2 for the cylinder

Or more simply the sphere's volume is \frac{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**:

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?

Amazingly we get the same ratio of \frac{2}{3} between the sphere's and cylinder's surface area!

But the same idea does not work for the cone.