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The ratio of the areas of the triangle and rectangle below is $1:2$.

enter image description here

So why isn't the ratio of volumes of a cone and the smallest cylinder that contains it $1:2$? If each "slice" has a $1:2$ ratio then shouldn't the volumes have a $1:2$ ratio as opposed to a $1:3$ ratio?

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  • $\begingroup$ Because volume is not surface area. $\endgroup$ – Thomas Apr 13 '14 at 16:41
  • $\begingroup$ (to make my first comment less cryptical: the volume of the cone is obtained by rotating the surface area aroung the vertical axis and 'summing'. 'Summing' is however happening in a weighted fashion. The further you are away from the centre of rotation, the higher the contribution. Now the part of the figure which adds to the volume of the cone is concentrated near the center, while the complement has contributions which are further away from the center). $\endgroup$ – Thomas Apr 13 '14 at 17:10
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Think about:

In your triangle picture, the ratio of triangle width to square width at varying heights decreases linearly from $1$ to $0$ (as you go from bottom of the figure to top). However, the ratio of cone cross-sectional area to cylinder cross-sectional area goes from $1$ to $0$ quadratically (because of area being proportional to square of radius).

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