Find the volume of a regular square pyramid whose square base has edge 12 inches and whose height is 8 inches.

Step-by-step solution

1. 1

   Write the pyramid volume formula. It works for any base shape:

   $V=\frac{1}{3}Bh,$

   where $B$ is the base area and $h$ the perpendicular height from apex to base.

2. 2

   Compute the base area. The base is a square of edge $12$ in:

   $B=12^{2}=144\ \text{in}^2.$

3. 3

   Identify the height correctly. The height is $h=8$ in — the perpendicular distance from the apex down to the base plane, not the slant height along a triangular face. Using a slant height here would overstate the volume.

4. 4

   Multiply and take a third.

   $V=\frac{1}{3}(144)(8)=\frac{1152}{3}=384\ \text{in}^3.$

5. 5

   Check against the matching prism. A box on the same $12\times 12$ base with height $8$ would hold $144\times 8=1152\ \text{in}^3$, and the pyramid holds exactly one third of that: $384\ \text{in}^3$ ✓. Comparing with the prism is the quickest way to catch a missing or doubled factor of three.