Find the ratio of the volume of a cone and a cylinder of same radii and same heights.

Answer

To find the ratio of the volume of a cone to the volume of a cylinder when both have the same radius and height, we can follow these steps:

### Step 1: Write down the formulas for the volumes.
- The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
- The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]

### Step 2: Substitute the values for radius and height.
Since both the cone and cylinder have the same radius \( r \) and height \( h \), we can use these formulas directly.

### Step 3: Set up the ratio of the volumes.
To find the ratio of the volume of the cone to the volume of the cylinder, we can write:
\[ \text{Ratio} = \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{1}{3} \pi r^2 h}{\pi r^2 h} \]

### Step 4: Simplify the ratio.
In the ratio, \( \pi r^2 h \) in the numerator and denominator cancels out:
\[ \text{Ratio} = \frac{\frac{1}{3}}{1} = \frac{1}{3} \]

### Step 5: Express the ratio in the required format.
Thus, the ratio of the volume of the cone to the volume of the cylinder is:
\[ 1 : 3 \]

### Final Answer:
The required ratio of the volume of a cone to the volume of a cylinder of the same radius and height is \( 1 : 3 \).

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