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What is the volume of the largest right cylinder that can fit inside a closed rectangular box measuring $12$ inches by $10$ inches by $8$ inches?

I thought we assume the radius of the cylinder equals to half the maximum dimension in the rectangular prism. $r=\dfrac{1}{2}max(\{12,10,8\},h=min(2\cdot r,\{10,12,8\} \cap \{2r\}^C)$

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I am also not sure what the cross-section should look if were not to enclose the cylinder in a sphere.

I imagine we center the cylinder's circular cross-section on the middle of the rectangle, but I am not sure on what rectangular side is most optimal.

I am not sure whether $12$ refers to a length, width, or height.

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  • $\begingroup$ MathJax hint: use \times to get a multiplication x, so 12 \times 10 gives $12 \times 10$. You get the right shape and spacing. \cdot is preferred for multiplication of numbers or terms, but \times is preferred for dimensions, cross products, and similar. $\endgroup$
    – Ross Millikan
    Aug 13, 2022 at 2:16

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It doesn't matter whether $12$ is length, width, or height. Those are arbitrary axes. The important question is which direction is the axis of the cylinder along? Calculate the largest cylinder that fits in each of the three axes and pick the largest. It is unlikely that the optimum will be along the middle axis, the $10$, but strange things happen. What is the largest circle that fits in a $12 \times 8$ rectangle? How can you use this to find the largest cylinder that fits in your box with the axis in one direction?

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