Answer
The volume of a pyramid is derived from the formula of volume of the rectangular solid. In the same way, the volume of a cone is derived from the formula of volume of the cylinder.
Work Step by Step
The volume of the pyramid is one-third of the volume of the rectangular solid figure. The base and height of the pyramid equal the base and height of the rectangular solid.In the same way, the volume of the cone is one-third of the volume of the cylinder with the same base and same height as dimensions.
The volume,\[\left( {{V}_{R}} \right)\] of a rectangular solid with length l, width w, and height h is given by the formula.
\[\left( {{V}_{R}} \right)=lwh\]
Accordingly,Considering the dimensions l, w,and h; as length, width,and height which is actually the perpendicular distance from the top of the base. The volume of the pyramid will be as follows:
\[{{V}_{P}}=\frac{1}{3}lwh\]
On equating both the aforesaid equations, one can derive the relation between their volumes as follows:
\[{{V}_{P}}=\frac{1}{3}{{V}_{R}}\]
Now, the volume,\[\left( {{V}_{CY}} \right)\] of a right circular cylinder with height h and radius r is given by the formula
\[\left( {{V}_{CY}} \right)=\pi {{r}^{2}}h\]
Also, the volume,\[\left( {{V}_{CO}} \right)\] of a right circular cone with height h and radius r is given by the formula
\[\left( {{V}_{CO}} \right)=\frac{1}{3}\pi {{r}^{2}}h\]
On equating the aforesaid last two equations, one can derive the relation between their volumes as follows:
\[\left( {{V}_{CO}} \right)=\frac{1}{3}\left( {{V}_{CY}} \right)\]
From both the above the evaluations, one can conclude that the volume of a pyramid is derived from the formula of volume of the rectangular solid. In the same way, the volume of a cone is derived from the formula of volume of the cylinder
