Cavalieri's Principle

This is Cavalieri's principle: if two solids stand between the same pair of parallel planes and every cross-section taken at the same level has the same area, then the two solids have the same volume.

Think of a straight stack of coins and the same stack pushed sideways into a lean. Every cross-section is still a coin of the same area, so the volume is unchanged, even though the outline looks different and the surface area is not the same.

Each solid has the same cross-section all the way up, so its volume is cross-sectional area $\times$ height.

Cylinder: $10\times 9=90{\text{ cm}}^3$. Rectangular prism: $10\times 9=90{\text{ cm}}^3$.

This is Cavalieri's principle: two solids with the same height whose cross-sections have equal areas at every level have equal volumes, whatever shape those cross-sections are. So yes, both solids have a volume of $90{\text{ cm}}^3$.

Each solid has a constant cross-section, so volume = cross-sectional area $\times$ height.

First solid: $20\times 12=240{\text{ cm}}^3$. Second solid: $20\times 15=300{\text{ cm}}^3$.

Since $240{\text{ cm}}^3\ne 300{\text{ cm}}^3$, the volumes are not the same. Cavalieri's principle needs both conditions to hold: matching cross-sectional areas at every level and the same height. Here the heights differ, so the principle does not apply and the taller solid has the greater volume.

Volume = cross-sectional area $\times$ height, so each solid has a volume of $25\times 10=250{\text{ cm}}^3$.

By Cavalieri's principle, two solids of the same height whose cross-sections have equal areas at every level have equal volumes. It makes no difference that one cross-section is a square and the other a circle: only the area of each cross-section matters, not its shape. So both volumes are $250{\text{ cm}}^3$.

Each solid has a constant cross-section, so volume = cross-sectional area $\times$ height.

First solid: $10\times 9=90{\text{ cm}}^3$. Second solid: $15\times 9=135{\text{ cm}}^3$.

Since $90{\text{ cm}}^3\ne 135{\text{ cm}}^3$, they do not have the same volume. Cavalieri's principle only guarantees equal volumes when the cross-sectional areas match at every height. Here they differ, so the solid with the larger cross-section has the larger volume.

Both stacks are $15\text{cm}$ tall and each has a constant cross-section, so volume = cross-sectional area $\times$ height.

First stack: $40\times 15=600{\text{ cm}}^3$. Second stack: $50\times 15=750{\text{ cm}}^3$.

With the heights equal, Cavalieri's principle tells us the volumes can only be equal if the cross-sectional areas are equal too. They are not, so the stack with the larger cross-section has the larger volume: the $50{\text{ cm}}^2$ stack, by $750-600=150{\text{ cm}}^3$.