# The object with finite volume but infinite surface area

By Murray Bourne, 22 Mar 2018

Here's an interesting paradox. I have a hollow object which has infinite height. I can fill the object with paint, but if I try to cover the surface with that paint, there won't be enough.

How can this be?

## Gabriel's horn

According to some, the archangel Gabriel (shared by Judaism, Islam and Christianity) is expected to blow a horn to indicate the last days are upon us.

The mathematical paint paradox involves the volume and surface area of a 3D object resembling Gabriel's horn in this picture.

## Generating the object

Consider the hyperbola :

We choose the right-hand portion (to avoid the discontinuity at *x* = 0) and plot the graph starting at *x* = 1. We consider the (infinite) shaded area to the right of the dashed line, below the curve and above the *x*-axis.

We rotate that area in 3 dimensions around the *x*-axis and end up with an (infinitely long) horn-like object.

### Volume of the horn

Let's now find the volume of the object we've drawn. We use the formula for the volume of a solid of revolution:

where:

*a* and *b* are the lower and upper *x*-limits of the shape

*y* is the curve we rotate around the axis, in this case 1/*x*

So in this case, our volume will be as follows (I'm retaining *b* as the upper limit since we can't substitute ∞ - we'll need to consider limits instead):

As *b* gets larger and larger, the final fraction in our answer gets smaller and smaller, so the volume is a finite quantity, π.

In mathematical notation, we would write this as:

### Surface area

Next, let's consider the surface area of the horn.

The formula for the surface area of a solid of revolution is:

where

*a* and *b* are the lower and upper *x*-limits of the shape

*y* is the curve we rotate around the axis

*dy/dx * is the derivative of *y*

In this case, we have:

If we consider the value of the fraction

as *x* gets very large, we can see it will get very close to 1/*x *(but will be just a little bit bigger than it).

So we can conclude that:

Now as *b* gets larger and larger, this quantity also gets larger. That is, it is an infinitely large quantity.

Here's the graph of our two solutions (in the variable *b*) - the magenta (pink) one is the volume V = π (1 − 1/*b*), while the green one is the surface area, S.A. = 2π ln(b). We can see the volume is tending to a limit (π), while the surface area just keeps on getting bigger.

So if we try to paint the outer surface with our π liters of paint, it will quickly run out.

## What's going on?

So how can it be that we have a finite volume, but an infinite surface area?

One way of looking at it is to recall how a volume of solid of revolution is actually calculated using integral calculus. We are finding the sum of the volumes of an infinite number of disks, radius *y* (for different values of *x*). The disks have area

and infinitesimal height *dx*. So each disk has volume

The sum of an infinite number of such disks will converge, since this general sum converges:

### The dispute

The paradox about an object having finite volume but infinite surface area caused a lot of dispute regarding the nature of infinity among mathematicians of the 17th century, including Galileo and Wallis. Such paradoxes are a great way to get us thinking!

Sources:

Gabriel's Horn

Gabriel's Horn Geogebra example

Photo

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