Volume of a Pyramid

The cube shown in the figure illustrates the calculation of the volume of a pyramid. The opposite vertices of the cube are connected, and the lines of connection meet at the center $O$. This divided the cube into six congruent pyramids.

The volume of any one the six pyramids such as $OABCD$ equals one-sixth of the volume of the cube and, therefore, the volume of one pyramid equals the area of the base $ABCD$ times one-third of $PQ$.

Thus, in this special case of a right square pyramid whose altitude equals one-half the length of a side of the base, we see that the volume of the pyramid equals the area of the base times one-third the altitude. In other words, the volume of the pyramid equals one-third the volume of a prism of the same base and altitude.

Rule: The volume of the pyramid equals the area of the base times one-third the altitude, i.e. $V = \frac{1}{3}Ah$, $A$ being area of base.

Example:

Find the volume of a pyramid whose base is an equilateral triangle of side $1$m and whose height is $4$m.

Solution:

Since the base of pyramid is an equilateral triangle of side $1$m,

The area of the base $= \frac{{{a^2}\sqrt 3 }}{4} = \frac{{1 \times 1.732}}{4} = 0.43$ square m

The volume of the pyramid $= \frac{1}{3} \times {\text{ area of base }} \times {\text{ height}} = \frac{1}{3} \times 0.43 \times 4 = 0.58$ cubic m.