If the volume of a cone is equal to one-third the volume of a cylinder with the same base and height, explain why this relationship holds true.

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The relationship between the volumes of a cone and a cylinder with the same base and height can be understood through the formulas for their respective volumes.

1. **Volume of a Cylinder**: The volume $$ V_c $$ of a cylinder is given by the formula:
   $$
   V_c = \pi r^2 h
   $$
   where $$ r $$ is the radius of the base and $$ h $$ is the height of the cylinder.

2. **Volume of a Cone**: The volume $$ V_{cone} $$ of a cone is given by the formula:
   $$
   V_{cone} = \frac{1}{3} \pi r^2 h
   $$
   where $$ r $$ is the radius of the base and $$ h $$ is the height of the cone.

Now, if we compare the two volumes, we can see the relationship:

- The volume of the cone is one-third of the volume of the cylinder:
  $$
  V_{cone} = \frac{1}{3} V_c
  $$

This relationship holds true because of the geometric properties of the cone and the cylinder. The cone can be thought of as a solid that tapers smoothly from the base to a point (the apex), while the cylinder has a uniform cross-section throughout its height.

To visualize why the cone's volume is one-third that of the cylinder, consider the following:

- If you were to fill the cone with a liquid and pour it into the cylinder, you would find that it takes three full cones to fill the cylinder completely. This is a result of the way the volume is distributed in the two shapes.

- Another way to understand this is through the concept of integration or calculus, where the volume of the cone can be derived by integrating the area of circular cross-sections from the base to the apex, leading to the factor of one-third.

In summary, the relationship that the volume of a cone is one-third the volume of a cylinder with the same base and height arises from the geometric properties of these shapes and is mathematically represented in their volume formulas.
The volume of a cone is one-third the volume of a cylinder with the same base and height because the cone's volume formula includes a factor of one-third compared to the cylinder's formula. This relationship is due to the geometric properties of the shapes.

If the volume of a cone is equal to one-third the volume of a cylinder with the same base and height, explain why this relationship holds true.

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