how could you estimate the volume of the box if you didn't know anything except the volume of the cylinder inside it?

Step-by-step explanation:

Since the box contains a cylinder, the height of the box must be gt; = cylinder height

And the area at the bottom of the box is definitely larger than the area at the bottom of the cylinder.

The area at the bottom of the cylinder is PI r squared,

If the bottom of the box is a square, then the area of the bottom is 4r^2, whereas the bottom tends to be polygon, which is equivalent to cutting the circle into equal parts by circle cutting.

So the area at the bottom of the box is ，

It follows that the volume of the box is greater than the volume of the cylinder