# 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 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 × 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 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)?*

## Surface Area

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 (\frac{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**):

Same Area

(Research "Archimedes' Hat-Box Theorem" to learn more.)