Definition: Cavalieri's principle says that two solids of equal height have equal volume if every parallel cross-section at the same height has equal area.
What is the idea?
Imagine two stacks of coins. One is a neat cylinder. The other is pushed sideways into a slanted tower. The coins are identical and the stack height is the same, so the volumes are equal, even though the shapes differ.
How is it used?
It justifies the volume formulas for oblique prisms, cylinders, pyramids, and cones. The volume of a slanted prism is still base area times perpendicular height, not times the slant length.
Example. A right cylinder has radius 5 cm and height 12 cm, so V = π(25)(12) = 300π. An oblique cylinder has the same radius, the same perpendicular height of 12 cm, and a slant edge of 15 cm. Every horizontal slice is a circle of area 25π, so its volume is also 300π, about 942.5 cubic centimeters.
What mistakes occur?
Using the slant length as the height. The height must be measured perpendicular to the base. Another error is to compare solids whose cross-sections are not equal at every level; the principle needs equal areas at every height, not just at the base.