Volume of Prisms and Pyramids

The volume of a prism is calculated by finding the area of its base and multiplying it by its height.
$Volume = Area \ of \ Base * Height$
The base can be any shape (square, hexagon, etc.). The area of that specific shape is used in the calculation.

A pyramid with the same base as a prism has less volume because it comes to a point, effectively shaving off volume.
The volume of a pyramid is one-third the volume of a prism with the same base and height.
$Volume \ of \ Pyramid = \frac{1}{3} * Area \ of \ Base * Height$
Important to use the perpendicular height (altitude), not the slant height.
Forgetting to multiply by $\frac{1}{3}$ will result in calculating the volume of the entire prism (box).

Consider a right triangle as the base of the pyramid.
A perpendicular line extends straight up from the base (imagine the corner of a room).
Given distances: base = 3, height = 4, pyramid height = 5 (all perpendicular to each other)
The base is a 3-4-5 right triangle. Note that 3-4-5 is a Pythagorean triple.
Area of the triangular base: $\frac{1}{2} * base * height = \frac{1}{2} * 3 * 4 = 6$
The height of the pyramid is 5 (perpendicular to the base).
Volume of the pyramid: $\frac{1}{3} * Area \ of \ Base * Height = \frac{1}{3} * 6 * 5 = 10$
Therefore, the volume of the triangular-based pyramid is 10 cubic units.