September 24, 2023

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These drinking glasses are too small!

Some of my reinsurance and math teacher friends may remember that when I’m out on the town and having an adult drink with friends, I’ve been known to stare at the drinking glass and say something like, “I mean Not to be rude, but the glasses here are definitely small. They’re much smaller than the ones we have at home. In fact, they’re so small that I think the circumference of the top of the glass is bigger than the height.”

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Then there is usually a long pause as the group considers it. The reinsurance group may need a few reminders of what perimeter means.

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Any group (unless they’ve heard it before, or unless they can guess it’s a setup) is likely to disagree with me. I’ll answer that I’m pretty sure of that, and I’m willing to bet a dollar.

How do you measure it in a bar or restaurant? I use a paper or cloth napkin to measure the circumference from one end of the napkin to somewhere in the middle of the napkin, and then I use that length to compare the height.

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I’ve done this enough times that I’m almost always right. Try it from your drinking glass. It consistently fails with a glass of champagne as well.

Recently it occurred to me that there should be a website that has a variety of glasses and their sizes, and I found, is a database of drawings with standard measurements. Measurements are based on industry standards and averages and may vary between manufacturers and regions. Here is a sample of the glasses with their images and measurements used here with permission, plus my calculations in the last three columns of the table. Volumes are in ounces, height and diameter are in centimeters. Source is and

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Here’s some R code to do the calculations and build the above graph. Note that pi (lower case) is a built-in R constant with a value of approximately 3.141593. (Yes, I’m well aware that π is an infinite, non-repeating decimal, and I believe R has 16 decimal digits, but that’s beyond the scope of this article.)

df volume = c(10, 22, 12.3, 25.36, 1.5, 2, 9),
height = c(11, 17.75, 19.8, 22.9, 15.9, 10.5, 23.5),
diameter = c (8, 9.5, 7.9, 10.8, 5.1, 4.13, 6.35))
df$circumference df$large df$height, “circumference”, “height”)
df$c_to_h df

library (ggplot2)
ggplot(df, aes(x=factor(glass), level=c(“Kalina 10”, “Pokal 22”, “Chardonnay”, “XL Oversize”, “cordial”, “shooter”, “Champagne”)), y = c_to_h , fill = glass, color = “black”)) +
geom_col(width = 1, position = position_dodge(1)) +
gom_hline (yintercept = 1) +
ggtitle(“ratio of perimeter to height by glass”) + xlab(“glass”) + ylab(“ratio”) +
theme(plot.title = element_text(face = “bold”, size = 12)) +
theme(axis.text.x = element_text(face = “bold”, size = 12)) +
subject(axis.text.y = element_text(size = 12, face = “bold”)) +
subject(legend. position = “none”) +
scale_fill_manual(“glass”, value = c(“red”, “yellow”, “blue”, “green”, “gray”, “brown”, “purple”))

I think it’s a good bet because the mind can’t easily compare a circular length to a linear length (I don’t know if this is scientifically accurate), plus maybe we look at the diameter but we forget We’re comparing height not to diameter, but to π times the diameter.

Feel free to place this bet with your friends or your students. How about sharing 10% of your winnings with me as commission?

Incidentally, a drinking vessel is a nearly right circular cylinder. (But not exactly; see above and below to see why.) Students of calculus can derive that the cylinder with the largest volume for a given surface area (the surface area can be represented as the rectangular region around a paper label) can be considered) the whole can), has a height equal to the diameter. The height of a typical 12 ounce soda may not equal the diameter, but its circumference is greater than its height. A fun supermarket experiment is to test different sized cans (a soup can, a tuna fish can, etc.) to determine which one meets the criteria for the largest volume.

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To my reinsurance friends: I learned the trick of height over girth from Paul Hawkesworth of M&G.


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