Volume Comparison of a Cylinder and a Cone with the Same Radius

When comparing the volumes of a cylinder and a cone with the same radius, a key observation is that the cone is half as tall as the cylinder. To explore this further, let's use the formulas for the volumes of these shapes and analyze the relationship between their volumes.

Formulas for Volume

The volume of a cylinder is given by the formula:

V_{cylinder} pi r^2 h

where ( r ) is the radius and ( h ) is the height of the cylinder.

The volume of a cone is given by the formula:

V_{cone} frac{1}{3} pi r^2 hend{latex}

where ( r ) is the radius and ( h ) is the height of the cone.

Given Data and Calculations

Suppose the radius ( r ) is the same for both the cylinder and the cone. The height of the cylinder is ( h ). Consequently, the height of the cone is ( frac{h}{2} ), which is half the height of the cylinder.

Calculating the Volumes

Volume of the cylinder:

V_{cylinder} pi r^2 h

Volume of the cone:

V_{cone} frac{1}{3} pi r^2 left(frac{h}{2}right) frac{1}{3} pi r^2 cdot frac{h}{2} frac{1}{6} pi r^2 h end{latex>

Comparing the Volumes

To compare the volumes directly, we can use the following ratio:

frac{ V_{cylinder}}{ V_{cone}} frac{pi r^2 h}{frac{1}{6} pi r^2 h} 6 end{latex>

Therefore, the volume of the cylinder is 6 times larger than the volume of the cone.

Conclusion

The mathematical comparison clearly indicates that for a given radius, if the cylinder is twice as tall as the cone, the volume of the cylinder will be six times the volume of the cone.

Additional Insights

Let's define some additional terms for clarity:

The volume of a cylinder is ( pi r^2 h ). The volume of a cone with the same radius and half the height is ( frac{1}{6} pi r^2 h ). The ratio of the volume of the cylinder to the volume of the cone is 6:1.

This ratio, 6:1, indicates that the cylinder, despite having the same radius as the cone, has more than double the base area when multiplied by its height, resulting in a volume that is six times larger.

Conclusion

From the derived formulas and the comparison, we have conclusively shown that the volume of the cylinder is six times greater than the volume of the cone when the radius is the same and the height of the cone is half that of the cylinder. This comparison underscores the significance of height in determining the capacity of these geometric shapes.