Gabriel’s Horn: A Shape with an Infinite Surface Area but a Finite Volume

I have seen posts before about an object that has an infinite surface area but a finite volume. The “wow” line for these posts is to note that there would be a finite amount of paint needed to fill the object but an infinite amount of painted needed to paint the object.^[1] Seems contradictory, doesn’t it?

There are several pages and posts on the internet that talk about this object. However, I want to give my own take on it.

First of all, what is this peculiar object with an infinite surface area and a finite volume? It is a Gabriel’s Horn. Here is my simple reconstruction of what a Gabriel’s Horn looks like:

This object certainly looks like a horn. Another appropriate way to describe it would be to call it a funnel. A funnel with a very long tip. In fact, the tip of the funnel is infinitely long. Note that in the illustration above, the tip of the funnel goes a little past 18 on the x-axis. The funnel stops there only out of convenience: in reality, the horn continues infinitely, continuing without limit.

To understand how Gabriel’s Horn has an infinite area but a finite volume, we will need to know how Gabriel’s Horn is created mathematically.  To create my Gabriel’s Horn, I began with the function

Then, I limited the function to the domain [1,∞). That is, I only let the graph exist starting at x = 1 and then I let the graph continue to the right to infinity. Note that the graph of this function does exist to the left of x = 1, but I did not need that part of the graph, so I limited myself to x = 1 to infinity. To help understand what I am talking about, below is the entire graph of the function

The part highlighted in red is the part that I actually used to create Gabriel’s Horn. I didn’t need the part in blue. Note that the transition from blue to red occurs where x = 1. So I took everything on the graph from one to infinity.

Once I took the limited part of the function, I spun it around the x-axis. In truth, I didn’t really spin it around the x-axis, but that is how I visualized Gabriel’s Horn. Once the limited function is spun around the x-axis, it would look a lot like the illustration shown previously: a funnel with an infinite length.

I think that is enough information to begin addressing the statement, “Gabriel’s Horn can be filled with a finite amount of paint but would require an infinite amount of paint to paint it.” My first observation is that Gabriel’s Horn exists in concept, not in reality. Sure, we can sketch what Gabriel’s Horn looks like, but even our sketches are limited: we do not actually see the end of the horn.

To give you an idea of how Gabriel’s Horn continues on and on to infinity, let us begin with our original sketch:

Now, let us magnify the funnel so that we can see it in more detail:

Note that the horn still has its proper proportions: one unit along the x-axis is the same length as one unit along the y-axis. Now, let us exaggerate the x-axis: let us compress the x-axis, while keeping the y-axis exactly as it is. Doing so gives us

The x-axis is approximately 20 times as long as the y-axis in the previous illustration, so the proportions are no longer correct, but it helps us see what happens as we get further to the tip of the funnel. Note that the funnel gets progressively narrower the further we go. And remember, the funnel goes to infinity. The closer we get to infinity, the narrower the funnel gets, but the funnel never actually pinches out of existence. It just gets thinner.

While it is possible to describe Gabriel’s Horn mathematically and to provide illustrations that show what it looks like, it would be impossible to create it in the real world. Why? Several reasons. Among them, the funnel would be much too fragile the thinner it gets. But most significantly, we would not be able to make an object that is infinitely long. It is simply impossible.

Thus, part of the way to address filling Gabriel’s Horn with paint versus painting Gabriel’s Horn is simply to note that, painting and filling with paint are real world activities. Since Gabriel’s Horn cannot exist in reality, it is meaningless to talk about filling it with paint or painting its surface. This is not to say that volume and surface area have no meaning to Gabriel’s Horn: they do. However, they can only be addressed mathematically, not in practice. Put another way, Gabriel’s Horn only has a conceptual volume and surface area. Trying to paint it or fill it up has no meaning.

Let us try a different approach. Even though we have already seen that a Gabriel’s Horn cannot exist in reality, let us suppose, for the sake of argument, that a real Gabriel’s Horn was made. How can it be painted? Well, paint the wide side of the funnel first, and then take your brush and begin at the wide side and do one continuous stroke until you reach the end of the funnel. When will you reach the funnel? Never. Your stroke will never come to an end because the end continues to infinity. Now, let us imagine filling the funnel with paint. We set the funnel upright (wide end up) and pour paint into its mouth. The paint then begins to fall its way down into the funnel.^[2] How long will it take the paint to reach the end of the funnel? It will never reach the end of the funnel. Since the funnel has an infinite length, it will take an infinite amount of time for a finite process (the paint falling down the funnel) to reach the end. Thus, even though the funnel has a finite volume, it would take an infinite amount of time to fill that finite volume. That just goes to show that we can only talk about the volume and surface area of Gabriel’s Horn as mathematical concepts. Those concepts do not translate into the real world.

If we cannot paint and fill a Gabriel’s Horn in practice, then how do we know its volume and surface area? We have to use calculus. Now, before you run away screaming from your computer or phone, let me assure you that the calculus we will be doing is easy. Also, the concept behind the calculus is not too difficult, either. So bear with me and follow along.

Recall that the Gabriel’s Horn was created by taking a graph of

and spinning it around the x-axis. We will come back to the idea of “spinning” something around the x-axis is just a moment. For now, let us visually cut the Gabriel’s Horn into thinner and thinner sections.

Here is the funnel end of Gabriel’s Horn.

Let us cut the horn at x = 1.5. It now looks like the following.

Let us cut it again, at x = 1.25. Now it looks like the following.

And once more, cutting it at x = 1.125, which gives us the following.

The section keeps getting thinner and thinner. What is the thinnest it can get? Well, we do not want the section to get so thin that it becomes zero, so we will use dx to represent the thinnest possible width before reaching zero.

How can we find the volume of this incredibly thin section? We can think of this section as a disc. Thus, it has a radius of 1 and a thickness of dx. Thus, the volume of that single, very first thinnest section is

Now, 𝜋dx doesn’t really have a meaning. After all, dx is the smallest number before zero, which isn’t really definable. However, what we can do is expand our equation to encompass every single thinnest section on Gabriel’s Horn. Since Gabriel’s Horn is

spun around the x-axis, the radius will always be 1/x. Thus, the volume of any thin section is

given any value for x. The volume of the whole horn is thus the sum of all of these thinnest of sections. Here is where calculus comes in: using calculus, we can set up an equation that gives us the sum of all of these sections from x = 1 to infinity. This equation is

To solve this equation (which is called a definite integral, by the way), we need to find an equation whose derivative is x^-2𝜋 and then solve for that equation at infinity and at 1, subtracting the two answers. The derivative is an equation that gives the slope of a graph at any point on the graph. The derivative can be calculated using the following formula:

If you do not understand why we do this, that’s okay. That is just how calculus works. However, I do want to point out two things. First, much of calculus is learning what are the derivatives of other equations. Once we know the various derivatives, simple calculus problems become simple. Second, calculus is not a more difficult math than algebra. Rather, it simply focuses on sequences that continue to infinity or that continue to zero. It is not more difficult: it is just a different subject.

Anyway, here is how we solve the previous equation:

Note that there is one peculiar part of the previous solution: dividing 1 by infinity. We just have to understand that, if infinity is a number increasing without end, then its inverse (that is, dividing one by infinity) is decreasing to zero. Technically, dividing by infinity does not make the number zero, but it is close enough to zero that we treat it as such.

This volume will be in cubic units, so the volume of Gabriel’s Horn is 𝜋 units³. Note that this is a finite volume.

Now, what about the surface area? We will start with the same place: the thinnest of thin sections. But now, how do we find the surface area? Let us consider that very first thin section, the one with a radius of one. The one shown below.

Rather than thinking of this as a disc, we will now think of it as a ring. A ring with a radius of 1 and a thickness of dx. Thus, its surface area will be the circumference of the ring times dx, or

Once again, we can translate this into an equation for any one of the thinnest sections by using a radius of

which gives us

We set this up as a definite integral from x = 1 to infinity and solve for the integral, which gives us

Now, there are a couple of peculiar calculations in the previous solution. First, in case you are wondering why the equation changed so radically, it is simply that the derivative of ln x is x^-1. It looks very different from before because the derivative of x^-1 is –x^-2. I do not want to go into detail here as to why the derivatives are so different, just understand that they are.

Second, taking the natural logarithm of infinity is technically not possible, but as the x-value on a natural logarithm graph approaches infinity, then the y-value also approaches infinity. Also, infinity times any number (in this case, times 𝜋) is infinity.

The answer is in square units, so the surface area of the Gabriel’s Horn is infinity units². Thus, the surface area of the horn is infinite.

Gabriel’s Horn kind of sits on a boundary. You can create a similar shape but using the equation

Here is a quick illustration of this new horn.

Because it is based on the square of the x, it has both a finite volume and surface area. On the other hand, we can produce another horn using the equation

This horn looks like the following.

Because it is based on the square root of x (or, using exponents, on the 1/2 power of x), it has both an infinite volume and surface area. As was already said, Gabriel’s Horn just happens to sit at a boundary of similar shapes but with larger or smaller exponents, and those other shapes have either an infinite volume and surface area or a finite volume and surface area.

Thoughts from Steven

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^[1]For example, https://www.reddit.com/r/wikipedia/comments/22t2s1/gabriels_horn_is_a_geometric_figure_which_has/

^[2]Technically, since the funnel is infinitely long, it would pass through the Earth and through the universe without reaching its end. Thus, the idea that gravity from the Earth is pulling the paint down to the end of the funnel has no meaning in reality. However, we are already allowing for the possibility of Gabriel’s Horn existing in reality, so let us go ahead and stipulate that paint will fall down the funnel, even if there would technically be no gravity actually pulling it down once the funnel extends past the Earth.