We are asked to describe how the volume of a cone and the volume of a cylinder are related. Let's then remember how we calculate each volume. We will start with the formula for the volume of a cone.
Volume of a Cone
The volume of a cone V is one third the area of the base B multiplied by the height h.
Since the base is a circle, we can use the radius to find its area.
We can use the formula for the area of a circle with radius r to find the area of the base B.
B = π r^2
Let's then substitute this expression into the formula for the volume.
V = 1/3Bh ⇒ V = 1/3(π r^2)h
The result is a formula for calculating the volume of a cone with radius r and height h.
Volume of a Cylinder
To calculate the volume of a cylinder, we multiply the area of the base by the height of the cylinder.
Same we noted earlier, if the radius of the circle base of a cylinder equals r, then the area of the base B is π r^2. We can substitute this expression into the formula for the volume.
V = Bh ⇒ V=(π r^2)h
The result is a formula for the volume of a cylinder with a radius r and height h.
Comparison
Note that the rules to calculating the volumes of these two figures are similar. In both cases we multiply the area of the base by the height of the figure. The only difference is that the volume of the cone is one third this product.
\begin{gathered}
V_\text{cylinder} = Bh \quad \text{ vs } \quad V_\text{cone} =\frac{1}{3}Bh
\end{gathered}
Note that if a cylinder and a cone had the same height and base area, the volume of the cone would be exactly one third the volume of the cylinder.
