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The volume of a cone is one third the product of its base area and its height.
The base area B is the area of the circle and the height h is measured perpendicular to the base.
V = 1/3 B h
Since the base is a circle, its area depends on its radius. Therefore, the base area can also be expressed in terms of the radius r.
V = 1/3 π r^2 h
A cone can be modeled as a stack of cylinders. The sum of the volumes of the small cylinders will be greater than the cone's volume. However, the higher the number of cylinders, the more the sum will approximate the volume of the cone.
Furthermore, the ratio of the sum of the volumes of each small cylinder to the volume of the big cylinder nears 13 as the number of stacked cylinders increases.
| Number of Cylinders | Sum of Volumes of Stacked Cylinders/Volume of Large Cylinder |
|---|---|
| 4 | ≈ 0.469 |
| 16 | ≈ 0.365 |
| 64 | ≈ 0.341 |
| 256 | ≈ 0.335 |
| 1024 | ≈ 0.334 |
| 4096 | ≈ 0.333 |
| ∞ | 1/3 |
Therefore, the volume of a cone is one third the volume of the cylinder with the same base area and height. \begin{gathered} V_\text{cone} = \dfrac{1}{3}V_\text{cylinder}\\[0.7em] \Downarrow \\ V_\text{cone} = \dfrac{1}{3} {\color{#FD9000}{Bh}} \end{gathered} Since the base area is the area of a circle, that formula can be substituted for B to find a more detailed formula for the volume of the cone. \begin{gathered} V_\text{cone} = \dfrac{1}{3}\pi r^2 h \end{gathered}