Slicing a solid with a plane shows a flat cross section; you sliced solids in Slicing Solids. Cavalieri's principle in Cavalieri's Principle is built on those slices. Going the other way, spinning a flat shape around a line sweeps out a solid, called a solid of revolution.
| Solid | Slice parallel to the base | Slice through the axis (vertical) |
|---|---|---|
| prism or cylinder | a copy of the base | a rectangle |
| pyramid | a smaller copy of the base | a triangle |
| cone | a circle | a triangle |
| sphere | a circle (any slice) | a circle |
Why it works
Each point of a spinning shape traces a circle around the axis. So a slice of a solid of revolution, perpendicular to the axis, is always a circle or a ring. A rectangle spun around one side sweeps a cylinder: the side on the axis becomes the height, and the other side becomes the radius. A right triangle spun around a leg sweeps a cone, and a semicircle spun around its diameter sweeps a sphere.
Key idea
Cross sections perpendicular to the axis of a prism, cylinder or pyramid copy (or shrink) the base. Spinning a rectangle, a right triangle or a semicircle around a side or diameter gives a cylinder, a cone or a sphere.
Worked example: Naming a cross section
A cube is sliced by a plane through three corners that are each one edge away from the same corner. What shape is the cross section?
- The three cut points are joined by diagonals of three faces of the cube, and all face diagonals are equal.
Answer: An equilateral triangle.
Worked example: Spinning a rectangle
A $4$ cm by $7$ cm rectangle spins around its $7$ cm side. Find the volume of the solid.
- The solid is a cylinder: the side on the axis ($7$) is the height and the other side ($4$) is the radius.
- $V = \pi \cdot 4^2 \cdot 7 = 112\pi \approx 351.9$ cm$^3$.
Answer: $112\pi$ cm$^3$
Worked example: Spinning a right triangle
A right triangle with legs $3$ and $6$ spins around the leg of length $6$. Find the volume.
- It sweeps a cone with height $6$ (the axis) and radius $3$.
- $V = \frac{1}{3}\pi \cdot 9 \cdot 6 = 18\pi$. (Around the other leg it would be $\frac{1}{3}\pi \cdot 36 \cdot 3 = 36\pi$: the choice of axis matters.)
Answer: $18\pi$
Common mistake
Mixing up which side becomes the radius. The side lying on the axis stays put and becomes the height; the side sticking out from the axis sweeps around and becomes the radius. Check with a quick sketch of the spun shape.