What is the relationship between the volumes of a cylinder and a cone, if they have the same heights and areas of their bases?

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To find the relationship between the volumes of a cylinder and a cone with the same heights and areas of their bases, we can use the formulas for the volumes of both shapes.

1. **Volume of a Cylinder (V_c)**:
   The volume $$ V_c $$ of a cylinder is given by the formula:
   $$
   V_c = A_b \cdot h
   $$
   where $$ A_b $$ is the area of the base and $$ h $$ is the height.

2. **Volume of a Cone (V_{cone})**:
   The volume $$ V_{cone} $$ of a cone is given by the formula:
   $$
   V_{cone} = \frac{1}{3} A_b \cdot h
   $$
   where $$ A_b $$ is the area of the base and $$ h $$ is the height.

3. **Relationship**:
   Since both the cylinder and the cone have the same base area $$ A_b $$ and height $$ h $$, we can express the relationship between their volumes as follows:
   $$
   V_c = A_b \cdot h
   $$
   $$
   V_{cone} = \frac{1}{3} A_b \cdot h
   $$

To find the ratio of the volumes, we can divide the volume of the cylinder by the volume of the cone:
   $$
   \text{Ratio} = \frac{V_c}{V_{cone}} = \frac{A_b \cdot h}{\frac{1}{3} A_b \cdot h}
   $$

Simplifying this gives:
   $$
   \text{Ratio} = \frac{A_b \cdot h}{\frac{1}{3} A_b \cdot h} = 3
   $$

Thus, the volume of the cylinder is three times the volume of the cone when they have the same height and base area.

In conclusion, the relationship between the volumes is:
$$
V_c = 3 \cdot V_{cone}
$$
The volume of the cylinder is three times the volume of the cone when they have the same height and base area.

What is the relationship between the volumes of a cylinder and a cone, if they have the same heights and areas of their bases?

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