Why is volume of cylinder > volume of cube If we have a cylinder with radius 1 and height 1 and cube with side lengths equal to 1
Volume of cube = $r^3 = 1^3 = 1$
Volume of cylinder = $\pi r^2h = \pi 1^2\times1 = \pi$
clearly $\pi > 1$, but if you think about it if you draw a square with side length one and a circle in the square, the circle has less area ( r = 1) e.g.
http://www.kevinhouston.net/blog/wp-content/uploads/2011/03/circle-in-square.png
So does this not imply that if you raise the two shapes the same height the square would be able to hold more water because the cylunder would fit into the square and there would still be space left over?
 A: if your cylinder is radius 1, then the diameter is 2, so it should be bigger then a cube with side length 1. A cylinder radius 1/2 would fit inside a cube. This cylinder would have a volume of $\pi(\frac 1 2)^2\times1 = \frac \pi 4$, which is smaller than 1.
A: Food for thought
I think many problems like this compare apples with oranges.  Let's put the cylinder and cube on the same scale by considering their volumes given the same surface area $S$.  The surface area of the cylinder of radius $r$ and height $h$ is
$$2 \pi r h + 2 \pi r^2$$
Because we have two parameters to vary, let's consider the cylinder of maximum volume given surface area $S$.  Then the volume is
$$V(r) = \frac12 S r - \pi r^3$$
We find the maximum by solving $V'(r)=0$ and noting $V''(r)<0$ and get that the max volume is achieved when $r=\sqrt{S/(6 \pi)}$.  Doing the algebra out, i.e., solving for $h$ and computing the volume, I get
$$V_{\text{cyl}} = \frac16 \sqrt{\frac{2}{3 \pi}} S^{3/2}$$
This is the maximum volume of a cylinder of given surface area $S$, which you may verify, by the way, has a square cross-section.  The cube is more straightforward:
$$V_{\text{cube}} = \left ( \frac{S}{6}\right)^{3/2}$$
The ratio is then
$$\frac{V_{\text{cyl}}}{V_{\text{cube}}} = \frac{2}{\sqrt{\pi}} \gt 1$$
which I hope now answers the question.
A: Say, I have a rectangular piece of paper 16x4. I make it into cylinder h=4 first, by connecting edges with a tape, but not overlapping. Then I bang the walls for each to be 4 inch, turning structure into rectangular form. That can not change the volume of the cylinder into some different volume.
A: Your cube formula is wrong. The volume of a cube that fits a cylinder with radius $r$ in it is $(2r)^3=8r^3$.
