Volume of Pyramids, Cones, and Spheres
Apply V = ⅓Bh for pyramids, V = ⅓πr²h for cones, and V = 4/3πr³ for spheres. Understand why pyramids and cones are one-third the volume of their corresponding prisms and cylinders (via Cavalieri's principle).
A lesson of Geometry, in Area and Volume, about 10 minutes.
By the end of this lesson, you will be able to find the volumes of pyramids, cones, and spheres and understand why the $\frac{1}{3}$ factor appears for pyramids and cones.
The Big Idea
Pyramids and cones have $\frac{1}{3}$ the volume of their corresponding prism or cylinder with the same base and height. This factor is not a coincidence — it follows from Cavalieri's principle and can be demonstrated experimentally by filling. The sphere formula is the most famous, derived by Archimedes.
Pyramids
$$V = \frac{1}{3}Bh$$
$B$ = area of base, $h$ = perpendicular height (from apex to base).
Why $\frac{1}{3}$? Three congruent square pyramids can be assembled into a cube. Volume of each = $\frac{1}{3}$ the cube. This generalizes to any pyramid via Cavalieri's principle.
Cones
$$V = \frac{1}{3}\pi r^2 h$$
Same $\frac{1}{3}$ factor — a cone is $\frac{1}{3}$ the volume of the cylinder with same base and height.
Physical demonstration: a cone filled with water and poured into a cylinder of same base and height fills it exactly $\frac{1}{3}$ of the way.
Spheres
$$V = \frac{4}{3}\pi r^3$$
Derived by Archimedes: a sphere of radius $r$ has $\frac{2}{3}$ the volume of its circumscribed cylinder (radius $r$, height $2r$).
$$V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \implies V_{\text{sphere}} = \frac{2}{3}(2\pi r^3) = \frac{4}{3}\pi r^3$$
Example 1 — Square Pyramid
Square base with side 6, height 8.
$$B = 36, \quad V = \frac{1}{3}(36)(8) = 96 \text{ cubic units}$$
Example 2 — Triangular Pyramid
Triangular base: base 5, height 4 (of triangle); pyramid height = 9.
$$B = \frac{1}{2}(5)(4) = 10, \quad V = \frac{1}{3}(10)(9) = 30 \text{ cubic units}$$
Example 3 — Cone
Radius 6, height 10.
$$V = \frac{1}{3}\pi(36)(10) = 120\pi \approx 376.9 \text{ cubic units}$$
Example 4 — Sphere
Sphere of radius 9.
$$V = \frac{4}{3}\pi(729) = 972\pi \approx 3053.6 \text{ cubic units}$$
Example 5 — Composite Solid
A cylinder of radius 5 and height 8 has a cone of radius 5 and height 3 on top. Find the total volume.
$$V_{\text{cylinder}} = \pi(25)(8) = 200\pi$$
$$V_{\text{cone}} = \frac{1}{3}\pi(25)(3) = 25\pi$$
$$V_{\text{total}} = 225\pi \approx 706.9 \text{ cubic units}$$