Cavalieri's principle

If the cross sections of two 2D objects at each height have the same length, the areas of the 2D objects are the same.

[asy] label("Equal Area:",(2.25,3)); draw((0,0)--(1,0)--(1.5,2)--(0.5,2)--cycle); draw("$1$",(0.25,1)--(1.25,1),grey); draw((3,0)--(4,0)--(4,2)--(3,2)--cycle); draw("$1$",(3,1)--(4,1),grey); [/asy]

If the cross sections of two 3D objects at each height have the same area, the volumes of the 3D objects are the same.

[asy] size(300); import three;  currentprojection=perspective(1,0,0.5);  label("Equal Volume:",(1.5,2));  pen bg=paleblue+opacity(0.2); draw(unitcone,bg); pen bg2=gray(0.9)+opacity(0.4); draw(unitcircle3, bg2);  draw(shift(-sqrt(pi)/2,3,0)*scale(sqrt(pi),sqrt(pi),1)*surface((0,0,0)--(1,0,0)--(0.5,0.5,1)--cycle),bg); draw(shift(-sqrt(pi)/2,3,0)*scale(sqrt(pi),sqrt(pi),1)*surface((1,1,0)--(1,0,0)--(0.5,0.5,1)--cycle),bg); draw(shift(-sqrt(pi)/2,3,0)*scale(sqrt(pi),sqrt(pi),1)*surface((1,1,0)--(0,1,0)--(0.5,0.5,1)--cycle),bg); draw(shift(-sqrt(pi)/2,3,0)*scale(sqrt(pi),sqrt(pi),1)*surface((0,0,0)--(0,1,0)--(0.5,0.5,1)--cycle),bg); draw(shift(-sqrt(pi)/2,3,0)*scale(sqrt(pi),sqrt(pi),1)*unitsquare3,bg2);  label("$\pi$",(0,0,0),bg2); label("$\pi$",(0,3+sqrt(pi)/2,0)); [/asy]

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