# 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.