Gabriel’s Horn

The area of this region is infinite. Here’s a proof:

The volume of this solid is finite. Here’s a proof:

So Gabriel’s Horn is a mathematical figure which has a finite volume (π), but which casts an infinite shadow!

If you find that this paradox challenges your faith in mathematics, remember that a cube with sides of length 0.01 casts a shadow that is 100 times as big as its volume. Gabriel’s Horn is just an infinite extension of this less paradoxical phenomenon.

r Consider a curve rotated about the x-axis: The surface area of this band is: r The radius is the y-value of the function, so the whole area is given by: This is the same ds that we had in the “length of curve” formula, so the formula becomes: Surface Area about x-axis (Cartesian):

The surface area of Gabriel’s Horn is infinite. Here’s a proof:

This means that if the horn was a paint can, it would not be able to hold enough paint to cover its own surface. However, this is seemingly impossible; think of it this way. Since the can would be full, the inside surface of the can (which, of course is the same as the outside surface, since the thickness of the can is 0) would be covered by paint. However, this area is so large that no amount of paint could possible cover it.