I enjoy a good pint of beer (or root beer, for my younger readers) as much as anyone. Which is why I was intrigued when I saw this article: http://www.avclub.com/denver/articles/of-legal-gauge,27021/ It’s about a guy who, tired of getting short poured at his local drinking establishment, created a gauge to measure the volume of beer in his pint glass based on the height of the beer. Reading the article made me curious about how one would determine the volume of beer in one’s pint glass.

As I write this, my pint glass is this full:

*Enjoy responsibly.
*

The casual observer might look at the glass and say, “Three quarters full! Oh yeah!” But are you ready to have your mind blown? Here it is: That glass is about half full. That’s right. Half. The reason has to do with the shape of the glass: it’s conic. Since the top of the glass is wider, it holds more beer than the bottom of the glass.

I guess the logical place to start here is to calculate the volume of beer in the pint glass. There are a couple ways to do it.* In the interest of being a responsible citizen, I’m going to use calculus because if you know calculus you’re probably at least close to legal drinking age. If you don’t know calculus, you should skip down to the paragraph that starts with “If you don’t know calculus, rejoin me here.”**

The shape of the pint glass lends itself to cylindrical coordinates, so let’s do a volume integral in cylindrical coordinates. We’ll assume the edges of the glass are straight, i.e. the glass is the shape of a cone with the point chopped off. I’ll draw half the glass on the cylindrical coordinate axes. I’m only drawing half because in the integral I’m going to integrate it from 0 to 2π (alternatively, I could draw the whole thing and only integrate from 0 to π):

In case it’s not clear from the drawing, r_{top} is the inner radius of the glass at the top, and r_{bot} is the inner radius of the glass at the bottom. L is the length of the part of the glass that can be filled.

Recall the general formula for a volume integral in cylindrical coordinates:

We need a function for r, which is just a straight line:

Where z is the height of beer in the glass. I’ll skip a few steps here and just give you the solution to the integral.

If you don’t know calculus, rejoin me here. I’m giving you a handy formula that can be used at any watering hole to see how much beer you’re actually getting, assuming a conic pint glass: (Note: this formula is unitless, so whatever units you plug in are the units you’ll get out. You may want to convert to ounces or milliliters to get a meaningful number.)

(Fun fact: If you set z=L and rbot=0, you get the formula for volume of a cone of height L. Neat-o!)

So now that we have the equation, let’s plug and chug (pun intended). Let’s calculate the volume of my pint glass, which as far as I know could be identical to any other conic pint glass. I measured the following dimensions, which may not be exact but they’re pretty close:

Using the above formula, it looks like my glass has a volume of 29.57 cubic inches, or 16.4 fluid ounces. A pint is 16 oz, so this is close enough given the rough measurements that I took. If I solve the equation for several values of z and graph it, it looks like this:

See how the slope increases as the glass gets fuller? This tells me that the volume increases at a higher rate as the glass gets closer to full. In other words, the math tells us what we already know intuitively: most of the volume is at the top of the glass! That’s because cross-sectional area increases as a function of the radius squared (Area = pi r^2) Therefore, since radius increases linearly with height, cross-sectional area increases as a function of the height of beer squared! Therefore, missing an inch of beer at the top of your glass is a way bigger deal than if you were missing an inch of beer at the bottom of your glass. In fact, you can see from the chart that an inch of beer missing from the top means you’re missing a full 25% of your pint! So next time the bartender pours you a beer, make sure he pours it to the top. And that’s how math can make you a better beer drinker.

*An inch from the top – almost full? No, only ¾ of a pint!*

*You should also be able to derive a volume formula not using calculus but instead based on simple high school geometry. (V=1/3bh, and some sines and cosines) I haven’t done it, but I’d love to see the derivation if anybody does!

**If you want to learn calculus, check out http://www.khanacademy.org/#precalculus as a place to start. The site has excellent, completely free, pre-recorded lectures on pretty much the entire standard math curriculum from arithmetic all the way up to differential equations. Calculus was a revolutionary way of looking at the physical world when it was discovered. It is still amazing and extremely useful.