The height and slant height of a cone are 12 cm and 13 cm respectively. Find its volume.

Answer

To find the volume of the cone given its height and slant height, we can follow these steps:

### Step 1: Identify the given values
- Height (H) = 12 cm
- Slant height (L) = 13 cm

### Step 2: Use the Pythagorean theorem to find the radius (R)
The relationship between the radius (R), height (H), and slant height (L) of a cone can be expressed using the Pythagorean theorem:
\[ L^2 = R^2 + H^2 \]

Rearranging this gives:
\[ R^2 = L^2 - H^2 \]

### Step 3: Substitute the known values into the equation
Substituting the values we have:
\[ R^2 = 13^2 - 12^2 \]
\[ R^2 = 169 - 144 \]
\[ R^2 = 25 \]

### Step 4: Calculate the radius (R)
Taking the square root of both sides:
\[ R = \sqrt{25} \]
\[ R = 5 \, \text{cm} \]

### Step 5: Use the formula for the volume of a cone
The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi R^2 H \]

### Step 6: Substitute the values into the volume formula
Now substituting the values of R and H:
\[ V = \frac{1}{3} \pi (5^2)(12) \]
\[ V = \frac{1}{3} \pi (25)(12) \]
\[ V = \frac{1}{3} \pi (300) \]
\[ V = 100 \pi \, \text{cm}^3 \]

### Step 7: Approximate the volume
Using \( \pi \approx \frac{22}{7} \):
\[ V \approx 100 \times \frac{22}{7} \]
\[ V \approx \frac{2200}{7} \]
\[ V \approx 314.29 \, \text{cm}^3 \]

### Final Answer
The volume of the cone is approximately \( 314.29 \, \text{cm}^3 \) or exactly \( 100 \pi \, \text{cm}^3 \).

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