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How can you forebode that the answer shall be independent of the number of sides of the base of the polygon ?

Stewart, Clegg, Watson. *Calculus Early Transcendentals*, 2021 9th edition. page 370. Problem 24.
>24. Given a sphere with radius r, find the height of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid is a regular n-gon? (A regular n-gon is a polygon with n equal sides and angles.) (Use the fact that the volume of a pyramid is $\dfrac{1}3Ah$, where A is the area of the base.)
When I attempted this question, I thought I flubbed up, because my answer lacked the number of sides of the base of pyramid! Even after seeing the solution, I still can’t intuit why the answer is independent of the number of sides of the base of the pyramid. Before attempting any paperwork, how can a student forefeel that the answer shall be independent of the number of sides of the base of the polygon?
![](https://i.imgur.com/suLunKZ.jpeg)

Stewart, Clegg, Watson. *Calculus Early Transcendentals*, 2021 9th edition. page 370. Problem 24.
>24. Given a sphere with radius $r,$ find the height of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid is a regular $n$-gon? (A regular $n$-gon is a polygon with $n$ equal sides and angles.) (Use the fact that the volume of a pyramid is $\dfrac{1}3Ah$, where $A$ is the area of the base.)
When I attempted this question, I thought I flubbed up, because my answer lacked the number of sides of the base of pyramid! Even after seeing the solution, I still can’t intuit why the answer is independent of the number of sides of the base of the pyramid. Before attempting any paperwork, how can a student forefeel that the answer shall be independent of the number of sides of the base of the polygon?
![](https://i.imgur.com/suLunKZ.jpeg)
[]()

calculus