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I was doing a few problems involving frustums, specifically right circular frustums, and I kept getting the surface area wrong using a certain method that made sense logically. Here is my method:

Obviously, a frustum is the result of a cone being cut by a plane perpendicular to the base of the cone. Therefore, we can subtract the surface area of the smaller cone formed by this cut from the surface area of the larger cone. However, we need to add the base of the smaller cone back since it is part of the surface of the frustum. I got the answer wrong both times I used this method, but the difference between the correct answer and my answer (correct answer - my answer) was equal to the area of the base of the smaller cone both times.

Here is one of the problems: "A cone with radius 9 and height 12 is cut in two pieces by a plane parallel to the base of the cone such that the plane is 8 units from the base of the cone. Find the total surface area of each piece thus formed."

I found the surface area of the small cone to be $24\pi$, which was correct. Then, I found the surface area of the larger cone to be $216\pi$. Also, the area of the smaller base was $9\pi$, which was correct as well. So, I did $216\pi - 24\pi + 9\pi = 201\pi$. The correct answer was $210\pi = 201\pi + 9\pi$.

Am I supposed to be adding that smaller base area back twice? If so, why?

anirudh30three
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  • You need to add $9\pi$ two times, because when you subtracted $24\pi$ you subtracted $9\pi$ which was not included in $216\pi$. Then you need to add it back to compensate this. Then you need to add it again to include it because it is part of frustum surface. – Ivan Kaznacheyeu Jul 28 '22 at 08:06
  • When you cut some figure by line of length $l$ total perimeter increases by $2l$. – Ivan Kaznacheyeu Jul 28 '22 at 08:08
  • I think I understand, but let me just confirm. The $216\pi$ is the surface area of the larger cone. However, when we cut the cone, we're adding extra area that isn't accounted for. Therefore, to include that area, we need to add another $9\pi$. Is that correct? – anirudh30three Jul 28 '22 at 18:25
  • Generally, it is correct. When we cut cone $216\pi$, we make additional area $9\pi$ twice: in frustum and in cone $24\pi$. Then total area of frustum and cone $24\pi$ is $216\pi + 2\cdot 9\pi$, then total area of frustum is $216\pi + 2\cdot 9\pi-24\pi=210\pi$. – Ivan Kaznacheyeu Jul 29 '22 at 08:06

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