[Math Lair] Estimating the Circumference of a Glass

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[Image of a beer glass] [Image of a beer glass]

To the right and the left are pictures of one of the glasses in my house. This beer glass is the tallest glass that I have. Which is greater, the height of the glass or the circumference around the rim? Make an estimate before reading on. You may also want to try this with one of your own glasses.

Most likely you've decided that the height of the glass is greater than the circumference. Measuring the height and circumference of the glass, the height was slightly under 9½" (the exact measurement was 93/8") and the circumference was slightly larger than 9½" (the exact measurement was 95/8"). So, the circumference is in fact greater than the height!

[The height of the glass is slightly less than 9.5 inches] [The circumference of the glass is slightly greater than 9.5 inches]

Try this with some of the glasses in your house. You'll almost certainly find that the circumference is greater than the height, unless you have some very tall, narrow glasses in your house. Those of you who are of age might be able to use this knowledge to win several bar bets.

It is difficult to estimate circumferences properly because the circumference is difficult to visualize. Probably the best way to guess is to estimate the diameter and, since the circumference is π multiplied by the diameter, attempt to visualize a length slightly greater than three of those distances.

You may also be interested in estimating the area of the border of postcards.