Volume of pyramids and cones

The one-third rule V = ⅓Ah for pyramids and cones, finding unknown dimensions and composite solids.

One third

Pyramids and cones

$$V = \frac{1}{3}Ah \qquad V_{\text{cone}} = \frac{1}{3}\pi r^2 h$$

Cone vs cylinder

For the same radius and height, the cone's volume (solid) is always one third of the cylinder's (dashed).

Example 1●

A square pyramid has base $10$ cm and height $12$ cm. Find its volume.

$V = \tfrac{1}{3} \times 100 \times 12 = 400$ cm³

Example 2●●

A cone has volume $12\pi$ cm³ and radius $3$ cm. Find its height.

$\tfrac{1}{3}\pi(9)h = 12\pi$, so $3h = 12$.

$h = 4$ cm

Flashcards

Cone volume

$\tfrac{1}{3}\pi r^2 h$

Pyramid volume

$\tfrac{1}{3} \times$ base area $\times$ height

Height to use

Vertical, not slant

Practice

Square pyramid, base $10$, height $12$. Volume?
Cone volume $12\pi$, radius $3$. Height?
Cone, radius $3$, height $4$. Volume (2 d.p.)?
Cone, radius $6$, height $10$. Volume (2 d.p.)?
Rectangular pyramid, base $4 \times 6$, height $4$. Volume?
A cylinder holds $300$ mL. How many full cones with the same base and height would fill it?
A pyramid has volume $75$ cm³ and base area $25$ cm². Height?
A cone has slant height $13$ and radius $5$. Its vertical height?