A spherical tomato and a cylindrical portion of a cucumber have the same height and radius. Then they are chopped into slices of equal thickness, as shown above.
Comparing each slice of both kinds, which slice will have more lateral surface area of the peel?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
Yea, isn't that theorem amazing! I was really surprised the first time I came across it, and couldn't believe it was for real.
I don't understand why you made it so complicated. Since you're just comparing lateral surface areas, then the lateral surface area of a cylinder (even after chopping it up) is 2 π r h and the lateral surface area of the sphere (even after chopping it up) is 4 π r 2 . With h = 2 r , we can show that they are equal.
Log in to reply
I'm not just comparing the whole lateral surface area, but also showing that each slice of a tomato and cucumber cut has equal area of peels as long as the thickness is the same.
Log in to reply
The question only asked us to compare the lateral surface areas of the two fruits. Why do you need to show that "each slice of a tomato or cucumber cut has equal area of peels as long as the thickness is the same" as well? It's like asking you how many cats you own, and you answered "I have 6 cats, 3 dogs, 2 hamsters and 5 rabbits," there's a lot of information that I don't need to know or didn't ask for.
Log in to reply
@Pi Han Goh – Don't you find it amazing that each slice has got the same lateral surface area? If you're asking why I showed this thing, I would simply answer "I found it amazing." That's all. There's a lot in math that I'd like to explore, and this is merely one of the facts.
Log in to reply
@Worranat Pakornrat – No, I understand what you're doing but if you're taking the entire height of the sphere/cylinder into account, then there isn't a need to a use that theorem. To make your question more interesting (by forcing readers to apply that theorem), you should ask for one "lateral surface area" of equal height, instead of the total "lateral surface area". With that, we have no choice but to apply that theorem.
By chopping them up and adding up all the lateral surface areas defeats the purpose of using that theorem.
Log in to reply
@Pi Han Goh – As a matter of fact, I didn't ask for the total surface area. I did ask for each slice if you notice. (I already bolded it again to emphasize.)
Log in to reply
@Worranat Pakornrat – You just updated the question after I replied to you. Your previous version did not reflect what you said earlier.
Log in to reply
@Pi Han Goh – Well, sorry about that then. Any other suggestion?
Log in to reply
@Worranat Pakornrat – Well, you should say that you cut a singular equal (but random) thickness for both fruits and then ask to compare the lateral surface area of these the two fruits.
Choice 1: same (right answer)
Choice 2: cylinder > sphere
Choice 3: cylinder < sphere
Choice 4: there is insufficient information.
Even after you made your edit, your image is still misleading and I suggest that you change it to one red and one green instead of 6 red and 6 greens (unless of course if you intentional want to confuse the readers, then it's fine).
Log in to reply
@Pi Han Goh – My original fourth option was deleted though. Is there any way to add the choice back in?
Log in to reply
@Worranat Pakornrat – Just go to the moderation channel and ask Andrew or Calvin or Sandeep or whoever is in charge.
Unfortunately the shape of the red tomato in the picture does not remember a sphere what is confusing. Indeed if one consider other shapes the result would be quite different!
Log in to reply
Sorry, that's my best graphic skills. It's better to stick to the context as the figure "may not be drawn to scale."
Thank you for the review, and yes, I couldn't believe it at first either! ;)
Problem Loading...
Note Loading...
Set Loading...
Relevant wiki: Surface Area of a Sphere
According to Archimedes' Hat-Box Theorem , the lateral surface area of the sphere portion will equal to the lateral surface area of the cylinder with the same height as the portion and the same radius as the whole sphere.
Considering the image above, let θ be the angle the portion's rim makes with the vertical axis. In the image, θ turns out to be a and b for the 2 right triangles in the sphere's portion. Then the surface area (A) of the sphere's portion can be calculated as the integration of infinitesimal areas of lateral cylindrical layers.
For each layer, the circumference equals 2 π r ( sin θ ) . With an infinitesimal change, d θ , the height of each layer is approximately r ( sin ( d θ ) ) ≈ r ( d θ ) .
Hence, the surface area can be calculated as:
A = ∫ a b ( 2 π r ( sin ( θ ) ) ) ( r ) d θ = 2 π r 2 ∫ a b sin ( θ ) d θ = 2 π r 2 [ − cos ( θ ) ∣ a b ] = ( 2 π r ) r [ cos ( a ) − cos ( b ) ]
Since the thickness of each slice h is the difference between the vertical sides of both triangles, it can calculated as:
h = ( r × cos ( a ) ) − ( r × cos ( b ) ) = r [ cos ( a ) − cos ( b ) ]
Substituting this term in the previous equation, we will get:
A = ( 2 π r ) r [ ( cos ( a ) − cos ( b ) ) ] = 2 π r h
This is clearly the formula for the cylindrical cucumber slice's lateral surface area with radius r and h . As a result, each tomato slice will have the same lateral surface area as the cucumber slice as long as the slice's thickness is the same!
Hence, each slice of both kinds have the same lateral surface area.