With Cavalieri's Principle in hand, the volume formulas for pyramids, cones and spheres stop being facts to memorize and become things we can explain. You used some of them already in Volume of Spheres.
Why it works
Pyramids: a cube can be cut into three congruent square pyramids, each with a face of the cube as its base and the cube's edge as its height (all three share the corner opposite their bases). So each pyramid is $\frac{1}{3}$ of the prism with the same base and height, and Cavalieri's principle extends that to every pyramid: $V = \frac{1}{3}Bh$. Cones: a cone matches a pyramid with the same base area and height, slice for slice, so $V = \frac{1}{3}\pi r^2 h$.
Spheres: slice a hemisphere of radius $r$ at height $y$: the slice is a circle of area $\pi(r^2 - y^2)$. Slice a cylinder of radius $r$ and height $r$ with a cone removed (point down) at the same height: the slice is a ring of area $\pi r^2 - \pi y^2$. Equal at every height, so the hemisphere's volume is $\pi r^3 - \frac{1}{3}\pi r^3 = \frac{2}{3}\pi r^3$. Double it: $V = \frac{4}{3}\pi r^3$.
Spheres: slice a hemisphere of radius $r$ at height $y$: the slice is a circle of area $\pi(r^2 - y^2)$. Slice a cylinder of radius $r$ and height $r$ with a cone removed (point down) at the same height: the slice is a ring of area $\pi r^2 - \pi y^2$. Equal at every height, so the hemisphere's volume is $\pi r^3 - \frac{1}{3}\pi r^3 = \frac{2}{3}\pi r^3$. Double it: $V = \frac{4}{3}\pi r^3$.
Key idea
Pyramid and cone: $V = \frac{1}{3}Bh$ (for a cone, $B = \pi r^2$). Sphere: $V = \frac{4}{3}\pi r^3$. The height $h$ is measured straight up from the base to the tip.
Worked example: A pyramid
A pyramid has a $6$ m by $6$ m square base and a height of $10$ m. Find its volume.
- $B = 36$ m$^2$.
- $V = \frac{1}{3} \cdot 36 \cdot 10 = 120$ m$^3$.
Answer: $120$ m$^3$
Worked example: A cone from its slant height
A cone has radius $5$ cm and slant height $13$ cm. Find its volume.
- The height, the radius and the slant height form a right triangle: $h = \sqrt{13^2 - 5^2} = \sqrt{144} = 12$ cm.
- $V = \frac{1}{3}\pi \cdot 5^2 \cdot 12 = 100\pi \approx 314.2$ cm$^3$.
Answer: $100\pi$ cm$^3$
Worked example: A sphere
Find the volume of a sphere with diameter $12$ in.
- The radius is $6$, so $r^3 = 216$.
- $V = \frac{4}{3}\pi \cdot 216 = 288\pi \approx 904.8$ in$^3$.
Answer: $288\pi$ in$^3$
Common mistake
Using the slant height in place of the height. The slant height runs along the side of a cone or pyramid; the height goes straight up from the center of the base. In the cone example, $13$ is the slant height and $12$ is the height. Check: the height is always the shorter of the two.