You know the volume of a right cylinder, $V = \pi r^2 h$, from Volume of Cylinders. What about a cylinder that leans? Picture a stack of coins pushed sideways. Every coin is the same coin as before, so the stack holds the same amount of metal. That simple idea is Cavalieri's principle.
Why it works
Think of a solid as a stack of very thin slices. The volume is the sum of the slices' volumes, and each thin slice's volume is about its area times its thickness. If two solids have the same height and, at every height, slices of the same area, then every pair of matching slices has the same volume, so the totals are equal. Sliding the slices sideways (making the solid oblique) changes nothing about any slice.
Key idea
Cavalieri's principle: two solids with the same height and equal cross-sectional areas at every height have the same volume. So an oblique prism or cylinder has volume $V = Bh$, where $h$ is the perpendicular height.
Worked example: An oblique cylinder
A leaning cylinder has radius $4$ cm, vertical height $10$ cm and slanted side $12$ cm. Find its volume.
- Each horizontal slice is a circle of area $\pi \cdot 4^2 = 16\pi$, the same as in an upright cylinder of height $10$.
- By Cavalieri's principle, $V = 16\pi \cdot 10 = 160\pi \approx 502.7$ cm$^3$. The $12$ cm slant is not the height.
Answer: $160\pi$ cm$^3$
Worked example: An oblique prism
An oblique prism has a $5$ in by $3$ in rectangular base and a vertical height of $8$ in. Find its volume.
- Every horizontal slice is a $5 \times 3$ rectangle, area 15.
- $V = Bh = 15 \cdot 8 = 120$ in$^3$.
Answer: $120$ in$^3$
Worked example: Cones and pyramids
A cone and a pyramid have the same height $h$ and the same base area $B$. Explain why they have the same volume.
- At height $y$, each slice is a copy of the base scaled by $\frac{h - y}{h}$ (it shrinks steadily to the tip), so its area is $B\left(\frac{h - y}{h}\right)^2$ in both solids.
- Equal slice areas at every height and equal heights: by Cavalieri's principle, equal volumes.
Answer: So once we know a pyramid's volume, a cone with the same base area and height has the same one.
Common mistake
Using the slanted length as the height. Cavalieri's principle compares slices at the same height above the base, so the height is always measured straight up, perpendicular to the base. Check: an oblique solid's slant edge is always longer than its height.