On June 7, 2013, my wife and I joined a bus trip titled Exhibition of Paintings from State Pushkin Museum and Flower Festival Commemorative Park. We had lunch at a French restaurant whose building was a nationally designated Important Cultural Property. At the beginning of lunch, each of us was served a colored glass with water and ice cubes (see the photo below). Water was the one from the well of the restaurant. On the other side of our table, a couple, looking younger than us, took their seats. Let’s call the lady of the couple Hana and the gentleman Ichiro.
The glass with water and dessert of the lunch course
At some stage of the lunch course, Ichiro took the glass and asked Hana if she could guess how to calculate the volume of such a body. The shape of the body is called a truncated cone or conical frustum. The latter said, “Add a small cone to make it a large cone. Calculate the volume of the large cone and subtract that of the small cone from it.” Ichiro replied that it would be very cumbersome. Then, he said, “We can get the volume as the mean of volumes of three cylinders having the same height as the solid.”
His voice was so low for me to hear the radii of the three cylinders. However, from the movement of his fingers, I supposed those were:
(1) the radius of the top circle of the truncated cone,
(2) the radius of the bottom circle of it, and
(3) a kind of mean of the above two.
He also stated that we could obtain the formula by integrating the circular cross-section area over the height of the frustum.
I had never heard of the formula for the volume of the truncated cone. So, I thought Ichiro wonderful to have learned the equation and remembered it for some reason. On the other hand, I wondered why he said Hana's simple method was cumbersome. Probably, he was a math teacher or rather an enthusiast for math.
After returning home, I calculated the volume using Hana’s “cumbersome method.” The result showed that the third radius mentioned by Ichiro was the geometric mean of the radii of the top and bottom circles.
You can see Ref. 1 to learn the formula and the derivation of it. The explanation is in Japanese, but people who do not read Japanese could also easily follow equations by looking at a diagram on that page. The third method of derivation mentioned there uses the Pappus–Guldin theorem (also known as Pappus's centroid theorem [Ref. 2]; the second theorem is relevant here). So, it might be hard to understand this method if you have never heard of that theorem.
References
“円錐台 (Truncated Cone),” Wikipedia, Japanese edition (last edited August 7, 2021) [https://ja.wikipedia.org/wiki/円錐台].
“Pappus’s centroid theorem,” Wikipedia, English edistion (last edited December 25, 2021) [https://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem].
Note added to the revised version: This article first appeared at https://ideaisaac.blogspot.com/2013/06/the-couple-talks-about-volume-of.html on June 19, 2013, and has been one of my most viewed posts (1.62-k views). In this revised version, I improved my English and made minor changes.